transport properties of foods – g. d. sanauacos & z. b. maroulis

TransportProperties of FoodsGeorge D. SanauacosRutgers UniversityNew Brunswick, New Jerseyand National Technical University of AthensAthens, Greece

Zacharias B. MaroulisNational Technical University of AthensAthens, Greece

MARCEL DEKKER, INC. NEW YORK • BASEL

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To our wivesKatie G. Saravacos andRena Z. Maroulis

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Preface

The basic transport properties of momentum (flow), heat and mass are animportant part of the engineering properties of foods, which are essential in thedesign, operation, and control of food processes and processing equipment. Theyare also useful in the quantitative analysis and evaluation of food quality and foodsafety during processing, packaging, storage and distribution of foods. The engi-neering properties are receiving increasing attention recently due to the need formore efficient processes and equipment for high quality and convenient foodproducts, under strict environmental and economic constraints.

The fundamentals of transport properties were developed in chemical engi-neering for simple gases and liquids, based on molecular dynamics and thermody-namics. However, the complex structure of solid, semi-solid, and fluid foods pre-vents the direct use of molecular dynamics for the prediction of the transport prop-erties of foods. Thus, experimental measurements and empirical correlations areessential for the estimation of these important food properties.

The need for reliable experimental data on physical properties of foods, es-pecially on transport properties, was realized by the development of national andinternational research programs, like the European cooperative projects COST 90and COST 90 bis, which dealt with such properties as viscosity, thermal conduc-tivity, and mass diffusivity of foods. One outcome of these projects was the impor-tance of context (relevancy) of the measurement and sample conditions. This ex-plains the wide variation of the food transport properties, particularly massdiffusivity.

Statistical analysis of compiled literature data may yield general conclusionsand certain empirical "constants", which characterize the transport property (ther-mal conductivity or moisture diffusivity) of a given food or food class.

All transport properties are structure-sensitive at the three levels, i.e. mo-lecular, microstructural, and macrostructural. Correlation of food macrostructureto transport properties is relatively easy by means of measurements of density,porosity, and shrinkage. Correlation to molecular and microstructural (cellular)structure, although more fundamental, is difficult and requires further theoreticaland applied work before wider application in food systems.

The material of this book is arranged in a logical order: The introduction,Chapter 1, summarizes the contents of the book, emphasizing the need for a uni-fied approach to the transport properties based on certain general principles. Chap-ter 2 introduces the fundamental transport properties as applied to simple gasesand liquids. The three levels of food structure, molecular, micro- and macrostruc-

vi Preface

ture, as related to transport properties are reviewed in Chapter 3. A unified treat-ment of the rheological properties of fluid foods is presented in Chapter 4. Thetheory, measurement and experimental data of moisture diffusivity are discussedin Chapter 5, while a statistical treatment of the literature data on moisture diffu-sivity is presented in Chapter 6. The diffusion of solutes in food systems is dis-cussed in Chapter 7, with special reference to flavor retention and food packagingfilms and coatings. Thermal conductivity and thermal diffusivity are discussed inChapter 8. Finally, heat and mass transfer coefficients are treated together inChapter 9.

We wish to acknowledge the contributions and help of several persons toour efforts over the years to prepare and utilize the material used in this book: Ourcolleagues, associates, and graduate students D. Marinos-Kouris, A. Drouzas, C.Kiranoudis, M. Krokida, N. Panagiotou, N. Zogzas of the National Technical Uni-versity, Athens; M. Solberg, M. Karel, J. Kokini, K. Hayakawa, V. Karathanos, S.Marousis, K. Shah, and N. Papantonis of CAFT and Rutgers University; M.Bourne and A. Rao of Cornell University, Geneva, NY; A. Kostaropoulos of theAgricultural University of Athens; and V. Gekas of the Technical University ofCrete. We also appreciate the discussions with the members of the Europeangroups of cooperative projects COST 90 and COST 90 bis, especially R. Jowittand W. Spiess.

Special thanks are due to Dr. Magda Krokida for her substantial contribu-tions in compilation and statistical analysis of the extensive literature data ontransport properties of foods, and her continued help in preparing the illustrationsand typing the manuscript. Finally, we wish to thank the staff of the publisherMarcel Dekker, Inc., especially Maria Allegra and Theresa Dominick, for theirhelp and encouragement.

We hope that this book will help the efforts to develop and establish foodengineering as a basic discipline in the wide area of food science and technology.We welcome any comments and criticism from the readers. We regret any errorsin the text that may have escaped our attention.

GeorgeD. SaravacosZacharias B. Maroulis

Contents

Preface

1. Introduction 1

I. RHEOLOGICAL PROPERTIES 3II. THERMAL TRANSPORT PROPERTIES 3III. MASS TRANSPORT PROPERTIES 4

2. Transport Properties of Gases and Liquids 7

I. INTRODUCTION 7II. ANALOGIES OF TRANSPORT PROCESSES 8III. MOLECULAR BASIS OF TRANSPORT PROCESSES 9

A. Ideal Gases 9B. Thermodynamic Quantities 10C. Real Gases 12

IV. PREDICTION OF TRANSPORT PROPERTIES OF FLUIDS 14A. Real Gases 15B. Liquids 16C. Comparison of Liquid/Gas Transport Properties 18D. Gas Mixtures 19

V. TABLES AND DATA BANKS OF TRANSPORT PROPERTIES 19

3. Food Structure and Transport Properties 29

I. INTRODUCTION 29II. MOLECULAR STRUCTURE 29

A. Molecular Dynamics and Molecular Simulations 29

vii

viii Contents

B. Food Materials Science 30C. Phase Transitions 30D. Colloid and Surface Chemistry 31

III. FOOD MICROSTRUCTURE AND TRANSPORT PROPERTIES 32A. Examination of Food Microstructure 32B. Food Cells and Tissues 32C. Microstructure and Food Processing 34D. Microstructure and Mass Transfer 34

IV. FOOD MACROSTRUCTURE AND TRANSPORT PROPERTIES 36A. Definitions 36B. Food Macrostructure and Transport Properties 40C. Determination of Food Macrostructure 45D. Macrostructure of Model Foods 46E. Macrostructure of Fruit and Vegetable Materials 50

4. Rheological Properties of Fluid Foods 63

I. INTRODUCTION 63II. RHEOLOGICAL MODELS OF FLUID FOODS 66

A. Structure and Fluid Viscosity 66B. Non-Newtonian Fluids 68C. Effect of Temperature and Concentration 71D. Dynamic Viscosity 73

III. VISCOMETRIC MEASUREMENTS 74A. Viscometers 74B. Measurements on Fluid Foods 78

IV. RHEOLOGICAL DATA OF FLUID FOODS 79A. Edible Oils 79B. Aqueous Newtonian Foods 80C. Plant Biopolymer Solutions and Suspensions 85D. Cloudy Juices and Pulps 89E. Emulsions and Complex Suspensions 90

V. REGRESSION OF RHEOLOGICAL DATA OF FOODS 92A. Edible Oils 92B. Fruit and Vegetable Products 94C. Chocolate 100

Contents ix

5. Transport of Water in Food Materials 105

I. INTRODUCTION 105II. DIFFUSION OF WATER IN SOLIDS 106

A. Diffusion of Water in Polymers 107III. DETERMINATION OF MASS DIFFUSIVITY IN SOLIDS 109

A. Sorption Kinetics 110B. Permeability Methods 114C. Distribution of Diffusant 118D. Drying Methods 120E. Simplified Methods 123F. Simulation Method 124G. Numerical Methods 124H. Regular Regime Method 12 5I. Shrinkage Effect 126

IV. MOISTURE DIFFUSIVITY IN MODEL FOOD MATERIALS 127A. Effect of Measurement Method 127B. Effect of Gelatinization and Extrusion 13 0C. Effect of Sugars 133D. Effect of Proteins and Lipids 13 5E. Effect of Inert Particles 137F. Effect of Pressure 138G. Effect of Porosity 140H. Effect of Temperature 141I. Drying Mechanisms 143

V. WATER TRANSPORT IN FOODS 144A. Mechanisms of Water Transport 144B. Effective Moisture Diffusivity 145C. Water Transport in Cellular Foods 146D. Water Transport in Osmotic Dehydration 147E. Effect of Physical Structure 150F. Effect of Physical/Chemical Treatments 152G. Characteristic Moisture Diffusivities of Foods 155

6. Moisture Diffusivity Compilation of LiteratureData for Food Materials 163

I. INTRODUCTION 163II. DATA COMPILATION 164

x Contents

III. MOISTURE DIFFUSIVITY OF FOODS AS A FUNCTION OFMOISTURE CONTENT AND TEMPERATURE 197

7. Diffusivity and Permeability of Small Solutesin Food Systems 237

I. INTRODUCTION 237A. Diffusivity of Small Solutes 237B. Measurement of Diffusivity 239

II. DIFFUSIVITY IN FLUID FOODS 241A. Dilute Solutions 241B. Concentrated Solutions 242

III. DIFFUSION IN POLYMERS 243A. Diffusivity of Small Solutes in Polymers 244B. Glass Transition 246C. Clustering of Solutes in Polymers 247D. Prediction of Diffusivity 248

IV. DIFFUSION OF SOLUTES IN FOODS 251A. Diffusivity of Salts 251B. Diffusivity of Organic Components 252C. Volatile Flavor Retention 254D. Flavor Encapsulation 258

V. PERMEABILITY IN FOOD SYSTEMS 259A. Permeability 259B. Food Packaging Films 261C. Food Coatings 262D. Permeability/Diffusivity Relation 263

8. Thermal Conductivity and Diffusivity of Foods 269

I. INTRODUCTION 269II. MEASUREMENT OF THERMAL CONDUCTIVITY

AND DIFFUSIVITY 270A. Thermal Conductivity 270B. Thermal Diffusivity 273

III. THERMAL CONDUCTIVITY AND DIFFUSIVITYDATA OF FOODS 275

Contents xi

A. Unfrozen Foods 275B. Frozen Foods 276C. Analogy of Heat and Mass Diffusivity 276D. Empirical Rules 279

IV. MODELING OF THERMAL TRANSPORT PROPERTIES 280A. Composition Models 280B. Structural Models 283

V. COMPILATION OF THERMAL CONDUCTIVITY DATA OF FOODS 289VI. THERMAL CONDUCTIVITY OF FOODS AS A FUNCTION OF

MOISTURE CONTENT AND TEMPERATURE 326

9. Heat and Mass Transfer Coefficientsin Food Systems 359

I. INTRODUCTION 359II. HEAT TRANSFER COEFFICIENTS 360

A. Definitions 360B. Determination of Heat Transfer Coefficients 361C. General Correlations of the Heat Transfer Coefficient 362D. Simplified Equations for Air and Water 364

III. MASS TRANSFER COEFFICIENTS 364A. Definitions 364B. Determination of Mass Transfer Coefficients 365C. Empirical Correlations 366D. Theories of Mass Transfer 367

IV. HEAT TRANSFER COEFFICIENTS IN FOOD SYSTEMS 369A. Heat Transfer in Fluid Foods 369B. Heat Transfer in Canned Foods 371C. Evaporation of Fluid Foods 372D. Improvement of Heat/Mass Transfer 373

V. HEAT TRANSFER COEFFICIENTS IN FOOD PROCESSING:COMPILATION OF LITERATURE DATA 374

VI. MASS TRANSFER COEFFICIENTS IN FOOD PROCESSING:COMPILATION OF LITERATURE DATA 391

Appendix: Notation 403

Index 407

1Introduction

The transport properties of momentum (flow), heat and mass of unit opera-tions are an important part of the physical and engineering properties of foods,which are necessary for the quantitative analysis, design, and control of food proc-esses and food quality.

The transport of momentum (rheological properties) and heat (thermal con-ductivity) have received more attention in the past (Rao, 1999; Rahman, 1995).However, mass transport is getting more attention recently, due to its importanceto several traditional and new food processing operations (Saravacos, 1995).

The transport properties of gases and liquids have been studied extensivelyand they are a basic element in the design of chemical processes and processingequipment (Reid et al., 1987). The theoretical analysis and applications of trans-port phenomena have been advanced by a unified treatment of the three basictransport processes (Brodkey and Hershey, 1988). The adoption of transport phe-nomena in food systems is expected to advance the emerging field of food engi-neering (Gekas, 1992). However, foods are complex heterogeneous and sensitivematerials, mostly solids or semisolids, and application of the principles of trans-port phenomena requires sustained experimental and theoretical efforts.

Application of modern computer aided design (CAD) to food processing hasbeen limited by the lack of reliable transport data for the various food processesand food materials. Mathematical modeling and simulations have made consider-able progress, but the accuracy of the available scattered data is not adequate forquantitative applications. Of particular importance is the need for mass transportproperties (Saravacos and Kostaropoulos, 1995; 1996).

While analysis and computation of the transport properties of gases and liq-uids is based on molecular dynamics, experimental measurements are necessaryfor the food materials and food processing systems.

2 Chapter 1

Theoretical analysis and experimental techniques of mass diffusion in poly-meric materials, developed in polymer science (Vieth, 1991) are finding importantapplications in food materials science and in food process engineering. Moleculardynamics and molecular simulation techniques, developed for the prediction ofmass diffusion in polymer science (Theodorou, 1996), could conceivably be util-ized in food systems, although the complexity of foods would make such an effortvery difficult.

The transport properties are directly related to the microstructure of foodmaterials, but limited studies and applications have been reported in the literature(Aguilera and Stanley, 1999; Aguilera, 2000). Food microstructure plays a particu-larly important role in mass transfer at the cellular level, for example in fruits andvegetables during osmotic dehydration.

Food macrostructure has been used widely to analyze and model transportmechanisms, particularly mass diffusivity and thermal conductivity. Simple meas-urements of density, porosity and shrinkage can provide quantitative informationon the heat and mass transport properties in important food processing operations,such as dehydration and frying. A thorough analysis of the transport propertiesshould involve the momentum, heat and mass transport mechanisms at the mo-lecular, microstructural, and macrostructural levels. Such a unified analysis mightreveal any analogies among the three transport processes, which would be veryhelpful in prediction and empirical correlations of the properties, like the analogiesfor gases and liquids.

Reliable data on transport properties of foods are essential because of thevarious non-standardized methods used, and the variability of composition andstructure of food materials. An international effort to obtain standardized data ofrheological properties (viscosity), heat conductivity, and mass diffusivity wasmade in the European collaborative research projects COST 90 and COST 90bis(Jowitt et al., 1983; 1987). The viscosity and thermal conductivity of foods wereinvestigated in a U.S. Department of Agriculture (USDA) cooperative researchproject (Okos, 1987). Accurate and useful data were obtained for viscosity andthermal conductivity, but only limited mass diffusivity data were obtained, dem-onstrating that mass transport is a much more complicated process. An importantconclusion of these projects is relevancy, i.e. each property refers to a given set ofexperimental conditions and sample material.

A comprehensive treatment of the transport properties of foods should bebased on the transport at the molecular, microstructural and macrostructural levels,and should consider the available literature data in a generalized form of statisticalanalysis.

Introduction 3

I. RHEOLOGICAL PROPERTIES

Food rheology has been primarily concerned with food texture and foodquality. However, rheological data of fluid foods are essential in the analysis anddesign of important food processing operations, like pumping, heating and cool-ing, evaporation, and thermal processing (both in cans and aseptic processing).

Most fluid foods are non-Newtonian fluids, and empirical Theological dataare necessary (Rao, 1999). Statistical (regression) analysis of published rheologi-cal data can provide useful correlations for groups and typical fluid foods (seeChapter 4).

The effect of temperature on the viscosity of fluid foods appears to be re-lated to the molecular and microstrure of the material: High energies of activationfor flow (about 50 kJ/mol) are observed in concentrated aqueous sugar solutionsand fruit juices, while very low values (near 10 kJ/mol) characterize the highlynon-Newtonian (and viscous) fruit purees and pulps (Saravacos, 1970).

II. THERMAL TRANSPORT PROPERTIES

Thermal conductivity represents the basic thermal transport property, and itshows a wider variation than thermal diffusivity, which can be estimated accu-rately from the thermal conductivity. The thermal conductivity of fluid foods is aweak function of their composition, and simple empirical models can be used forits estimation. Structural models are needed to model the thermal conductivity ofsolid foods, which varies widely, due to differences in micro- and macrostructureof the heterogeneous materials. Heat and mass transfer analogies in porous foodsmay be related to the known analogies of gas systems.

Application of structural models of thermal conductivity to model foods hasdemonstrated the importance of porosity of granular or porous materials (Marouliset al., 1990). Regression analysis of published data of thermal conductivity ofvarious foods, as a function of moisture and temperature, can provide useful em-pirical parameters characteristic of each material. Such parameters are the thermalconductivity and the energy of activation of dry and infinitely wet materials (seeChapter 8).

Chapter 1

III. MASS TRANSPORT PROPERTIES

The diffusion model, developed for mass transport in fluid systems (Cussler,1997), has been applied widely to mass transfer in food materials, assuming thatthe driving force is a concentration gradient. Since mass transfer in heterogeneoussystems may involve other mechanisms than molecular diffusion, the estimatedmass transport property is an effective (or apparent) diffusivity. Most of the pub-lished data on mass transport in food systems refer to moisture (water) diffusivity(Marinos-Kouris and Maroulis, 1995), since the transport of water is of fundamen-tal importance to many food processes, like dehydration, and to food qualitychanges during storage.

Mass transport in foods is strongly affected by the molecular, micro- andmacrostructure of food materials. The crucial role of porosity in moisture transferhas been demonstrated by measurements on model foods of various structures, andon typical food materials (Marousis et al., 1991; Saravacos, 1995). The effect oftemperature on moisture diffusivity may provide an indication whether mass trans-fer is controlled by air or liquid/solid phase of the food material. Low energies ofactivation for diffusion (about 10 kJ/mol) are obtained in porous materials, whilehigh values (near 50 kJ/mol) are observed in nonporous products.

The wide range of moisture diffusivities reported in the literature is causedprimarily by the large differences in mass diffusivity among the vapor, liquid, andsolid phases present in heterogeneous food materials. The diffusivity in the solidphase is also affected strongly by the physical state, i.e. glassy, rubbery or crystal-line. Application of polymer science to food systems containing biopolymers canimprove the understanding of the underlying transport mechanisms (see Chapters5 and 7).

Statistical (regression) analysis of published literature data on moisture dif-fusivity, using an empirical model as a function of moisture content and tempera-ture, can provide useful parameters, such as diffusivity and activation energy inthe dry and infinitely wet phases (see Chapter 6).

Cellular models for mass transfer can provide an insight into the process ofosmotic dehydration, where water and solutes are transported simultaneously.However, the diffusion model is often used, because of its simplicity, for the esti-mation of mass diffusivity of water and solutes during the osmotic process.

Mass transport of important food solutes, such as nutrients and flavor/aromacomponents, is usually treated as a diffusion process, and effective mass diffusivi-ties are used in various food processes and food quality changes, like aroma reten-tion (see Chapter 7).

Introduction

REFERENCES

Aguilera, J.M. 2000. Microstructure and Food Product Engineering. Food Technol54(ll):56-65.

Aguilera, J.M., Stanley, D.W. 1999. Microstructural Principles of FoodProcessing, Engineering. 2nd ed. Gaithersburg, MD: Aspen Publ.

Brodkey, R.S., Hershey, H.C. 1988. Transport Phenomena. A Unified Approach.New York: McGraw-Hill.

Cussler, E.L. 1997. Diffusion Mass Transfer in Fluid Systems. Cambridge, UK:Cambridge University Press.

Gekas, V. 1992. Transport Phenomena of Foods and Biological Materials. NewYork: CRC Press.

Jowitt, R., Escher, F., Hallstrom, H., Meffert, H.F.Th., Spiess, W.E.L., Vos, G.,eds. 1983. Physical Properties of Foods. London: Applied Science Publ.

Jowitt, R., Escher, F., Kent, M., McKenna, B., Roques, M., eds. 1987. PhysicalProperties of Foods 2. London: Elsevier Applied Science.

Marinos-Kouris, D., Maroulis, Z.B. 1995. Transport Properties in the Air-Dryingof Solids. In: Handbook of Industrial Drying, 2nd ed. Vol.1, Mujumdar, A.S.ed. New York: Marcel Dekker.

Maroulis, Z.B., Drouzas, A.E., Saravacos, G.S. 1990. Modeling of ThermalConductivity of Granular Starches. J Food Eng 11:255-271.

Marousis, S.N., Karathanos, V.T., Saravacos, G.S. 1991. Effect of Physical Struc-ture of Starch Materials on Water Diffusivity. J Food Proc Preserv 15:183-195.

Okos, M.R., ed. 1987. Physical and Chemical Properties of Foods. ASAE Publica-tion No. Q0986, St. Joseph, MI.

Rahman, S. 1995. Food Properties Handbook. Boca Raton, FL: CRC Press.Rao, M.A. 1999. Rheology of Fluid and Semisolid Foods. Gaithersburg, MD: As-

pen Publ.Reid, R.C., Prausnitz, J.M., Poling, B.E. 1987. The Properties of Gases and Liq-

uids. 4th ed. New York: McGraw- Hill.Saravacos, G.D. 1970. Effect of Temperature on the Viscosity of Fruit Juices and

Purees. J Food Sci 35:122-125.Saravacos, G.D. 1995. Mass Transfer Properties of Foods. In: Engineering Proper-

ties of Foods. 2nd ed. Rao, M.A., Rizvi, S.S.H. eds. New York: Marcel Dek-ker, pp. 169-221.

Saravacos, G.D., Kostaropoulos, A.E. 1995. Transport Properties in Processing ofFruits and Vegetables. Food Technol 49(9):99-105.

Saravacos, G.D., Kostaropoulos, A.E. 1996. Engineering Properties in FoodProcessing Simulation. Computers Chem Engng 20:S461-S466.

Theodorou, D.N. 1996. Molecular Simulations of Sorption and Diffusion inAmorphous Polymers. In: Diffusion in Polymers. Neogi, P. ed. New York:Marcel Dekker, pp. 67-142.

Chapter 1

Vieth, W.R. 1991. Diffusion In and Through Polymers. Munich, Germany: HanserPubl.

Transport Properties of Gases andLiquids

I. INTRODUCTION

The physical processes and unit operations of process engineering are basedon the transport phenomena of momentum, heat, and mass (Bird et al., 1960;Geankoplis, 1993). The transport phenomena, originally developed in chemicalengineering, can be applied to the processes and unit operations of food engineer-ing (Gekas, 1992). The analogy of momentum, heat and mass transport facilitatesa unified mathematical treatment of the three fundamental transport processes(Brodkey and Hershey, 1988).

The transport properties of simple gases and liquids have been investigatedmore extensively than the corresponding properties of solids and semisolids. Mo-lecular dynamics and thermodynamics have been used to predict, correlate andevaluate the transport properties of simple gases and liquids (Reid et al., 1987).Empirical prediction methods, based on theoretical principles, have been used topredict the transport properties of dense gases and liquids, compiling tables anddata banks, which are utilized in process design and processing operations.

This chapter presents a review of the molecular and empirical prediction oftransport properties of gases and liquids, with examples of simple fluids of impor-tance to food systems, like air and water. The theoretical treatment of simple fluidsis useful in analyzing and evaluating the transport properties of complex food ma-terials.

Chapter 2

II. ANALOGIES OF TRANSPORT PROCESSES

The transport processes of momentum (fluid flow), heat and mass can beexpressed mathematically by analogous constitutive equations of the general form(one-dimensional transport):

<p = -S(dy/8x) (2-1)

where fx is the transport rate, 8 is the transport coefficient or property, anddy/dx is the transport gradient.

The negative sign of Eq. (2-1) denotes that the transport is in the ^-direction,i.e. the transport gradient (dy/dx) is negative (Brodkey and Hershey, 1988).

Equation (2-1) is a generalized expression of the empirical laws of Newton,Fourier and Pick for the transport of momentum, heat and mass, respectively:

r» = -T] (du/dx) (2-2)

(q/A\ = -A (dT/dx) (2-3)

(j/A)x = -D (dC/dx) (2-4)

The basic transport properties, defined by the last equations, are:77: viscosity, Pa s or kg/m sX: thermal conductivity, W/m KD: mass diffusivity, m2/s

In the transport equations, T = F/A is the shear stress in the y-direction for flowin the ^-direction (Pa), u is the velocity in the ^-direction (m/s), T is the tempera-ture (K), and C is the concentration (kg/m3). The following transport properties arealso used:

v = Tj/p: momentum diffusivity or kinematic viscosity (m2/s)

a = ^1 p cp : thermal diffusivity (m2/s)

where p (kg/m3) is the density and cp (J/kg K) is the heat capacity at constant pres-sure of the fluid.

Thus, all three transport properties, v, a, D are expressed in the same units,m2/s.

Transport Properties of Gases and Liquids 9

It should be noted that, although the three transport processes are expressedmathematically by the same generalized transport equation (2-1), the mechanismsof transport of momentum, heat and mass may be quite different.

III. MOLECULAR BASIS OF TRANSPORT PROCESSES

Molecular dynamics, which is concerned with intermolecular forces andmolecular movement, can be utilized in the prediction of transport properties ofsimple fluids. The mechanism of the three transport processes are different (Brod-key and Hershey, 1988). Thus, momentum transport is caused by the relative mo-tion of fluid layers parallel to the direction of flow; heat conduction is caused bycollision of the molecules, without substantial movement of the species; and massdiffusion is caused by movement of molecules in mixtures.

A. Ideal GasesThe ideal gases are considered to consist of rigid spherical molecules obey-

ing the laws of mechanics, and their transport properties can be predicted by thekinetic theory of gases. Although very few gases approach ideality (Ar, Xe, Kr),the concept of ideal gas is useful in understanding the transport properties.

The mean free path /lm of a gas, defined as the average distance of molecularmovement before collision with another molecule or surface, is given by the equa-tion (Brodkey and Hershey, 1988):

X =5/0= *>T ' (2-5)" ' ' n l

where u is the mean velocity (m/s), 9 is the collision frequency (1/s), d is the mo-lecular diameter (m), P is the pressure (Pa), and kB is the Boltzmann constant kB =1.38xlO'23J/moleculeK.

From Eq. (2-5) it follows that,

^mP = constant (2-6)

This means that the mean free path of the molecular motion is inversely propor-tional to the pressure.

10 Chapter 2

The transport properties of ideal gases, viscosity 77, thermal conductivity /I,and mass diffusivity D are given by the following equations:

(2-7)

(2-8)

D = - u Am (2-9)

Since v = r/jp and a = A//? cp , Eqs. (2-7) to (2-9) yield the following analogies:

), = cvi] = pcvD (2-10)

v = ay = D (2-11)

where y = cp /cy , and cp, cv are the heat capacities at constant pressure and con-stant volume, respectively.

B. Thermodynamic QuantitiesThe transport processes of momentum, heat, and mass take place in systems

that are removed from thermodynamic equilibrium. Thermodynamics cannot pre-dict transport properties, but some thermodynamic quantities are used in molecularand empirical predictions (Prausnitz et al., 1999).

Pressure-volume-temperature (PVT) data are needed in calculations and cor-relations of transport properties of fluids. PVT data are usually obtained from thecubic equations of state, such as the Redlich-Kwong equation of state (Smith andvan Ness, 1987):

V-b

Transport Properties of Gases and Liquids 1 1

Equation (2-12) yields the known cubic equation of van der Waals and it reducesto the ideal gas law (PV=RT), if the empirical constants a and b are taken equal tozero. The cubic equations of state can be transformed to polynomials of third de-gree in respect to volume V.

The empirical constants of the Redlich-Kwong equation are related to thecritical properties P0 Tc of the fluid:

(2-13)

^008664^ (2_M)

The critical properties of a fluid, Pc, Vc, and Tc, are characteristic for each fluid,and they are used to calculate the reduced properties:

pr = p/pc, v, = V/Vc, Tr = T/Tc (2-15)

The residual or excess properties of a fluid are the differences between the real andthe ideal properties. Thus, the residual volume VKS is defined as:

(2-16)

RTtherefore V = z—— (2-17)

where z = PV/RTis the compressibility factor, which expresses the non-idealityof the fluid.

The theorem of corresponding states indicates that all real gases, when com-pared at some reduced pressure and temperature, have the same compressibilityfactor z and all deviate from the ideal gas behavior to almost the same degree.

The molecular structure of real nonspherical gases is characterized by theacentric factor co, which is related to the molecular shape. For ideal gases ca = 0,and for real gases, the acentric factor is related to the reduced pressure Pr by theequation:

a> = 1.0-log(P,)rr.,7 (2-18)

For many fluids the normal boiling point is approximately equal to 0.7TC.

12 Chapter 2

Table 2.1 Critical Constants and Acentric Factors of Selected FluidsFluidOxygenNitrogenCarbon DioxideEthyleneEthanolWater

Vc , crrrVmol73.489.893.9

130.4167.155.1

rc ,K

154.6126.2304.1282.4531.9647.3

Pc,bar50.433.973.850.461.4

221.2

zc

0.2880.2900.2740.2800.2400.235

CO

0.0250.0390.2390.0890.6440.344

ource: Data from Reid et al., 1987.

Table 2.1 shows the critical constants V0 Tc, P0 zc and the acentric factor a>for selected fluids of interest to food systems.

C. Real GasesThe intermolecular forces of fluids constitute the basis of all transport proc-

esses at the molecular level. Their origin and determination is treated in the spe-cialized literature (Maitland et al., 1981). In real gases (monoatomic, polyatomic,nonpolar, and polar) the Chapman-Enskog theory of nonuniform gases is usuallyapplied. In the prediction of the transport properties, the following empirical pa-rameters are usually employed (Reid et al., 1987): collision diameter a, potentialenergy s, and collision integral Q.

The potential energy u(r) or the Lennard-Jones 12-6 potential of two spheri-cal nonpolar molecules is given by the equation:

u(r)=4s (2-19)

where cr is the characteristic collision diameter, similar to molecular diameter d ofthe kinetic theory, £ is the minimum value of u(r), and r is the intermolecular dis-tance.

The parameters cr and s are determined from empirical thermodynamicequations as functions of the critical pressure Pc, the critical temperature Tc and theacentric factor co of the fluid:

=1CT10(2.3551-0.00876;) (2-20)


Table 2.2 Intermolecular Constants of Selected ComponentsGasOxygenNitrogenAirCarbon DioxideEthyleneEthanolWater

a, nm0.350.380.370.390.420.450.26

£/kB,K106.771.478.6

195.2224.7362.6809.1

Source: Data from Reid et al., 1987. Inm = 10A = 1 0 m

s/kBTc =0.17915 + 0.169o> (2-21)

Table 2.2 shows the empirical constants crand e of gases important to foodsystems.

The collision integral Q of real fluids is a measure of the active cross sectionof collision, depending on temperature, which is related to the intermolecular po-tential by a complex integral (Assael et al., 1996). The collision cross section for amolecule is the area, perpendicular to the direction of movement, within which thecenter of a second molecule should be located in order to collide.

The collision integral is related to the transport properties of viscosity Qnand mass diffusivity £2D. The two integrals are different, and the collision integralfor viscosity Qn is usually estimated more accurately than QQ, since viscosity 77 isdetermined more accurately than mass diffusivity D. The following empiricalequation is used for estimation of fin (Brodkey and Hershey, 1988):

where T" = TkB/ sis the reduced temperature.The collision integral for diffusion f2D of nonpolar fluids is estimated as a

complex function of the reduced temperature T*.For polar fluids, the collision integral for diffusion is estimated from the

equation:

. /r (2-23)

where 5* is the dipole dimensionless number, defined by the equation:

14 Chapter 2

where (DPM) is the dipole moment (given in thermodynamic tables) and V0, Tbare the molar volume (m3/mol) at the normal boiling point (K). Water, a typicalpolar molecule, has a value DPM= 1.8 debye.

IV. PREDICTION OF TRANSPORT PROPERTIES OF FLUIDS

The transport properties of fluids can be considered that they consist of threecontributions, as shown in the following equation (Assael et al., 1996; Millet et al.,1996):

X(p,T)=X.(T) + AX(p,T) + A* (p,T) (2-25)

where X(p, T) is the transport property (TJ, /I, D), X0(T) is the transport property ofthe dilute fluid (gas), AX(p, T) is the excess contribution of the real fluid, andAXc(p,T) is the critical contribution. The terms X0(T)+ AX(p,T) represent the basicpart of the transport property, while the critical contribution AXc(p, T) becomes ofimportance near the critical temperature. The transport properties are affectedmainly by the temperature T and the density or concentration p, while pressuremay have an effect in some special (e.g. critical) conditions.

The excess contributions are important in predictions of viscosity (At] = 77-rjq) and thermal conductivity (AA = /L-Ag), and they can be estimated when somedata are available in the literature. The critical contribution is more important for Athan for 77.

The accuracy of prediction is higher for viscosity (1-3%) than for thermalconductivity (about 10%). The mass diffusivity is predicted with lower accuracy(10-50%), especially at high concentrations of the diffusant (Brodkey and Her-shey, 1988).


A. Real GasesThe transport properties of real gases, viscosity 77 (Pa s), thermal conductiv-

ity /I (W/m K), and mass diffusivity D (m2/s) can be predicted by the Chapman-Enskog equations, based on the intermolecular parameters (Brodkey and Hershey,1988):

= 2.669x10" (Mr)" (2-26)

= 8.3224x10' (2-27)

D = 1.883x10- (2-28)

where M is the molecular weight, T is the temperature (K), P is the pressure (Pa),a is the collision diameter (m) and /?7i QD are the collision integrals.

Equations (2-26) to (2-28) show that 77 and /I increase with the square rootof temperature, while D is a function of the cubic power of temperature. Equation(2-28) indicates that, at constant temperature, PD=constant, i.e. the mass diffusiv-ity is inversely proportional to the pressure.

Pressure has a negligible effect on /I of gases up to 10 bar, but it is importantat higher pressures, especially near the critical condition.

The viscosity 77 and the thermal conductivity /I of real gases are correlatedby empirical equations, which facilitate the interconversion of the two transportproperties. For monoatomic gases the Eucken factor is used:

AM = 2.5 (2-29)

where M is the molecular weight (kg/kmol) and cv is the heat capacity at constantvolume (kJ/kmol K).

For polyatomic gases, the Eucken factor is given by the equation:

AM , 2.25 , 2.25—— = 1 + ——— = 1 +cvIR c IR-\

(2-30)

16 Chapter 2

The heat capacities cv and cp are related by the equation:

cp=cy+R (2-31)

where R = 8.314 kJ/kmol K is the gas constant.

B. LiquidsIn liquids, the intermolecular forces are stronger than in gases, due to the

close proximity of the molecules. Prediction of the transport properties by molecu-lar parameters is difficult, and empirical relationships are normally used. Experi-mental measurements of the transport properties of liquids are necessary to vali-date the empirical prediction equations. Measurement techniques are discussed inthe treatment of transport properties of food materials (see Chapters 5 and 7).

/. ViscosityThe Eyring theory of rate processes yields an empirical expression for vis-

cosity, which is similar to the Arrhenius equation:

(2-32)

where A and B are empirical constants for the particular liquid.The effect of temperature is sharper at higher viscosities, i.e. the sensitivity

of viscosity to temperature variations depends primarily on the value of viscosity.The viscosity-temperature relationship for liquids is expressed by the empiricalLewis-Squires equation and diagram (Reid et al, 1987; Syncott, 1996):

-0,66, -0,66,

233

when 77, r/0 are the viscosities (Pa s) at temperatures T, T0, respectively.


2. Thermal ConductivityThe thermal conductivity of liquids can be estimated from empirical equa-

tions as a function of the temperature T, like the following expression:

/i = A + BT + CT2+DT3 (2-34)

where the constants A, B, C, D are given in tables (Reid et al., 1987).In most liquids, except water and some alcohols, /I is a negative function of

temperature. Pressure has a negligible effect on A up to 50-80 bar, but it becomesimportant near the critical point, where the gas behaves like a liquid.

3. Mass DiffusivityThe diffusivity of a species A in a liquid medium B can be estimated from

the Wilke-Chang equation (Brodkey and Hershey, 1988):

(2-35)

where rj is the viscosity of the liquid (Pa s), T is the temperature (K), MB is themolecular weight of B, VA is the molar volume of A at the boiling point (m3/kmol),and <f> is an interaction parameter, e.g. 2.26 for water and 1.5 for ethanol.

In general, the mass diffusivity of a particle in a liquid medium is given bythe Stokes - Einstein equation:

If TD = B (2-36)

where rp is the particle radius (m), TJB is the viscosity of the liquid medium (Pa s),T is the temperature (K), and &s=1.38xlO"23 J/molecule K is the Boltzmann con-stant.

From both Eqs. (2-35) and (2-36) it follows that

= constant (2-37)

i.e. the mass diffusivity is inversely proportional to the viscosity of the solution.

18 Chapter 2

The Eyring theory of rate processes predicts for mass diffusivity an Ar-rhenius-type relationship, analogous to viscosity:

D = Ae\p(-ED/RT) (2-38)

where A is a constant and ED is the energy of activation for diffusion (kJ/kmol).Experimental measurements of mass diffusivity in liquids are less accurate

than for viscosity, especially at high concentrations of the diffusant. Mass diffusiv-ity in aqueous systems is important in food and biological materials (Cussler,1997) and is treated in Chapter 7 of this book.

The mass diffusivity in electrolytes is affected strongly by the ionic speciesof the solution. In dilute solutions of a single salt the Nernst-Haskel equation isused to estimate the diffusivity D°B as a function of the temperature, the valencesof anions and cations, the ionic conductances, and the Faraday constant (Reid etal., 1987). For concentrated solutions, the empirical Gordon equation is used,which corrects the D°AB for viscosity, molality and ionic activity of the solute.

The mass diffusivity of a typical electrolyte, sodium chloride, goes througha minimum at normality 0.2 N (DAB=l .2\\0'9 m2/s at 18.5 °C), and it increases atlower and higher concentrations.

C. Comparison of Liquid/Gas Transport PropertiesBoth viscosity 77 and thermal conductivity /I of liquids are much higher than

the corresponding properties of gases. These differences reflect the stronger inter-molecular forces of the dense liquid state. However, the mass diffusivity in theliquid state is much lower than in dilute gases, due to the difficulty of mass trans-port in dense molecular systems. Selected values of transport properties of fluidsof importance to food systems are given in Tables 2.3-2.5. Typical values for airand water at 25 °C are the following:

• air: 77 = 0.017 mPas, A = 0.025 W/m K• water: r;=0.90mPas, A = 0.62W/mK

oxygen/air: DAB= 1.7x1 0"5m2/s•• oxygen/water: DAB- 1.7x1 0"9m2/s

For comparison of transport properties in the liquid (L), and gas (G) state,the following approximate ratios are useful:

tljjj0 = A t / A G = 10 to 1000, DJDG si/10000 (2-39)


D. Gas MixturesThe viscosity of steam/air mixtures, which is important in retort processing

of packaged foods, is given by the empirical equation (Kisaalita et al., 1986):

^=vayJ°y*+t?,y,Tby' (2-40)

where rja, rjs are the air and steam viscosities respectively, o=0.039, 6=0.0163,ya, ys are the mass fractions of air and steam, respectively, and Tis the tempera-

ture in °C.The calculated values of varied from 0.0144 Pa s (100 °C, yg= 0.1) to

0.0204 (140 °C, y, = 0.5).

V. TABLES AND DATA BANKS OF TRANSPORT PROPERTIES

The transport properties are usually included in tables and data banks ofphysical properties of materials. Data on transport properties of common fluids arefound in the well-known Perry's Chemical Engineers' Handbook (1984), in Reidet al. (1987) and in Cussler (1997). The extensive compilation of thermophysicalproperties of matter by Touloukian (1971) includes transport property data. Exten-sive compilation of physical properties, including transport properties of fluids, areavailable in computer databases and data banks, such as the Dechema databank ofthe German Chemical Society (Eckermann, 1983), and the PPDS data bank of theNational Engineering Laboratory, U.K. (PPDS, 1996).

The Design Institute for Physical Property Data (DIPPR) of the AmericanInstitute of Chemical Engineers (AIChE) recently published tables of transportproperties for binary mixtures (DIPPR, 1997) and a data bank of the same proper-ties (DIPMIX, 1997). Compilations of data and plots of physical properties of purecompounds are available for Windows (DIPPR, 1998).

Packages of extensive data banks of physical properties, requiring main-frame computers, are used in process equipment design and in processing opera-tions. Software of selected physical properties for desktop computers and Win-dows is available (CEP, 1999). Most of the data on transport properties of fluidspublished in tables and data banks are related to the chemical, petroleum, and pet-rochemical industries. Limited information is available on food and biologicalproducts, which are, for the most part, solid or semisolid materials.

20 Chapter 2

Tables 2.3 and 2.4 give some selected values of transport properties of sim-ple fluids of importance to food processing and food engineering. They are usefulin determining and evaluating the transport properties of food materials, which aretreated in subsequent chapters of this book.

More data for viscosity and thermal conductivity of air and water (liquid andvapor) as a function of temperature are presented in Figures 2.1 and 2.2.

Table 2.3 Viscosity 77 and Thermal Conductivity A ofSelected Gases and Liquids (25°C)

Material

GasesAirOxygenNitrogenCarbon dioxideEthyleneEthanolWater vaporLiquidsWaterEthanol

r|, mPa s

0.0170.0180.0180.0150.0120.0250.010

0.901.04

A., W/m K

0.0250.0200.0260.0160.0200.0150.020

0.620.15

Source: Data from Perry, 1984 and Reid et al., 1987.

Table 2.4 MassDiffusant (A)

OxygenNitrogenCarbon dioxideEthanolEthyleneWaterSalt (NaCl)

Diffusivity (DAB)in Air (B)

(atm pressure, 25°C)DAB, xlO'5m2/s

1.71.81.91.32.12.0

-

in Water (B)(dilute solutions, 25°C)

DAB, xlO'9m2/s1.71.92.01.31.91.11.2

Source: Data from Perry, 1984 and Reid et al., 1987.


BB.E

0.001

0.01

200

Temperature (°C)

300 400

Figure 2.1 Viscosity of air and water versus temperature. (Adapted from Pa-kowskietal., 1991.)

22 Chapter 2

•oeo

0.01200

Temperature (°C)

300 400

Figure 2.2 Thermal conductivity of air and water versus temperature. (Adapted fromPakowski et al., 1991.)


A third degree polynomial was fitted to these data and the results are sum-marized in Table 2.5. A different equation was used for viscosity data of water,since the results of the third degree polynomial are not adequate.

In the same table an empirical equation from Pakowski et al. (1991) is pre-sented for predicting the water vapor diffusivity in air versus temperature andpressure (see also Figure 2.3).

Table 2.5 Empirical Equations for Calculating the Transport Properties of Water,Water Vapor and Air

Saturated VaporSuperheated Vapor

Air

7.95E-03 4.49E-058.07E-03 4.04E-051.69E-02 4.98E-05

a2

-6.13E-081.24E-09

-3.19E-08

a3

1.44E-10-1.21E-121.32E-11

Temperature (°C)0-300

100-7000-1000

Saturated Waterao ai a2 a3 Temperature (°C)

-1.07E+01 1.97E-02 -1.47E-05 1.82E+03 0-350

Thermal Conductivity(W/m K)

Temperature (°C)Saturated WaterSaturated Vapor

Superheated VaporAir

5.70E-011.76E-021.77E-022.43E-02

1.78E-031.05E-046.01E-057.89E-05

-6.94E-06-6.7 IE-079.5 IE-08-1.79E-08

2.20E-093.07E-09-3.99E-11-8.57E-12

0-3500-300

100-7000-1000

Mass Diffusivitv(m2/s)

D=ao(T+273)/273)ai P"2

Water Vapor in Air 2.16E-05 1.80E+00 -l.OOEOOTemperature (°C)

0-1200

Source: Data from Pakowski et al. (1991)

24 Chapter 2

l.E-02

l.E-0610 100

Temperature (°C)

1000

Figure 2.3 Diffusivity of water vapor in air. (Data from Pakowski et al., 1991.)

Figure 2.4 shows the viscosity of aqueous sucrose solutions as a function ofconcentration and temperature. Figure 2.5 shows the thermal conductivity of com-mon gases of importance to food processes as a function of temperature.


percentage sucrose by weight = 60

50

Temperature (°C)

100

Figure 2.4 Viscosity of aqueous sucrose solutions. (Adapted from Perry et al.,1984.)

26 Chapter 2

0.070

0.060

0.050

•aae(J

0.040

0.030

0.020

0.010-200 200

Temperature (°C)

600

Figure 2.5 Thermal conductivity of gases. (Adapted from Perry et al., 1984.)


REFERENCES

Assael, M.I., Trusler, M.J.P., Tsolakis, T.F. 1996. Thermophysical Properties ofFluids. An Introduction to Their Prediction. London: Imperial College Press.

Bird, R.B., Stewart, W.E., Lightfoot, E.N. 1960. Transport Phenomena. NewYork: John Wiley & Sons.

Brodkey, R.S., Hershey, H.C. 1988. Transport Phenomena. A Unified Approach.New York: McGraw-Hill.

CEP - Chemical Engineering Progress. 1999. Software Directory. New York:AIChE.

Cussler, E.L. 1997. Diffusion Mass Transfer in Fluid Systems. 2nd ed. Cambridge:Cambridge University Press.

DIPMIX. 1997. Database of Transport Properties and Related ThermodynamicData of Binary Mixtures. College Station, TX: Engineering Research Station,Texas A&M University.

DIPPR. 1997. Transport Properties and Related Thermodynamic Data of BinaryMixtures. Volumes 1-5. New York: AIChE.

DIPPR. 1998. Data Compilation of Pure Compound Properties. New York: Tech-nical Data Services Inc.

Eckermann, R. 1983. Information systems for, and prediction of, physical proper-ties of non-food materials. In: Physical Properties of Foods. Jowitt, R.,Escher, F., Hallstrom, B., Meffert, H.F. Th., Spiess, W.E.L., Vos, G. eds.London: Applied Science Publ.

Geankoplis C.J. 1993. Transport Processes and Unit Operations. 3rd ed. NewYork: Prentice Hall.


Kisaalita, W.S., Lo, K.V., Staley, L.M. 1986. A Simplified Empirical Expressionfor Estimating the Viscosity of Steam/Air Mixtures. JFoodEng 5:123-133.

Maitland, G.C., Rigby, M., Smith, E.B., and Wakeman, W.A. 1981. Intermolecu-lar Forces. Their Origin and Determination. Oxford: Clarendon Press.

Millet, J., Dymond, J.H., Nieto de Castro, C.A. 1996. Transport Properties of Flu-ids. Their Correlation, Prediction and Estimation. Cambridge: CambridgeUniversity Press.

Pakowski, Z., Bartczak, Z., Strumillo, C., Stenstrom, S. 1991. Evaluation of Equa-tions Approximating Thermodynamic and Transport Properties of Water,Steam and Air for Use in CAD of Drying Processes. Drying Technol 9:753-773.

Perry R.J., Green, J.H., Moloney, J.O. 1984. Perry's Chemical Engineers' Hand-book. 6th ed. New York: McGraw-Hill.

PPDS-Physical Properties Data Service. 1996. Physical Properties Databases.N.E.L., East Kilbride, U.K.

Prausnitz, J.M., Lichtenthaler, R.N., Azevedo, E.G. 1999. Molecular Thermody-namics of Fluid Phase Equilibria. 3rd ed. New York: Prentice Hall.

28 Chapter 2

Reid, R.C., Prausnitz, J.M., Poling, B.E. 1987. The Physical Properties of Gasesand Liquids. 4th ed. New York: McGraw-Hill.

Smith, J.M., van Ness, B.C. 1987. Introduction to Chemical EngineeringThermodynamics. New York: McGraw-Hill.

Syncott, K. 1996. Chemical Engineering Design. In: Coulson and RichardsonChemical Engineering. Vol. 6. pp 274-280.

Touloukian. Y. S. 1971. Thermophysical Properties of Matter. Volumes 1-13.New York: IFI/ Plenum.

Food Structure and TransportProperties

I. INTRODUCTION

Food structure at the molecular, microscopic and macroscopic levels, usedin the study and evaluation of food texture and food quality, can be applied to theanalysis and correlation of the transport properties of foods. Molecular dynamicsand molecular simulations, recently developed in polymer science for the study ofpolymer structure, can be extended to more complex food systems, improving theempirical mechanisms and correlations of transport processes. Food macrostruc-ture, used in engineering and processing applications, can be related to the recentadvances of food microstructure at the cellular level. Applications of food struc-tural analysis and experimental data to the transport properties, especially masstransport, will improve food process and product development, and food productquality.

I. MOLECULAR STRUCTURE

A. Molecular Dynamics and Molecular SimulationsThe transport properties of simple gases and liquids can be predicted by

theoretical and semiempirical models based on molecular dynamics and thermo-dynamics. Extensive data on the physical and transport properties of gases andliquids are available in the form of tables and data banks, as outlined in Chapter 2

29

30 Chapter 3

(Reid et al., 1987). These correlations and data are of limited use to food systems,since food materials are generally heterogeneous solids, which are difficult to ana-lyze and interpret in terms of pure molecular science.

The mechanical and transport properties of polymers can be predicted by thenew technique of molecular simulation, which is based on molecular dynamicsand uses extensive computer computations. Molecular simulations use statisticalanalysis and computer computations of a particular material, from which struc-tural, thermodynamic and transport properties are estimated (Theodorou, 1996).

Molecular simulations can produce polymer configurations based on equi-librium statistical mechanics, which show the distribution of sizes and shapes ofopen spaces, formed within the polymer structure, where the penetrant moleculescan reside. Application of this technique to polymer science could produce specialpolymers for specific application, e.g. separation of various molecules.

B. Food Materials ScienceThe major components of foods are biopolymers (proteins, carbohydrates

and lipids) and water. Food materials science is essentially polymer science ap-plied to food materials. Polymer science was developed largely in the syntheticpolymer (plastics) industry, but recently it is applied at a growing rate in food sci-ence and technology (Slade and Levine, 1991; Levine and Slade, 1992).

The transport properties of foods are closely related to the properties of foodbiopolymers, as shown in the analysis of viscometric properties (see Chapter 4),mass diffusivity (see Chapters 5, 6 and 7) and thermal conductivity/diffusivity (seeChapter 8). The techniques of polymer science are used widely in the determina-tion of mass transport (diffusion) of water and other solutes in food materials(Vieth, 1991).

Food biopolymers of importance to transport properties are structural pro-teins (collagen, keratin, elastin), storage proteins (albumins, globulins, prolaminsand glutenins), structural polysaccharides (cellulose, hemicelluloses, pectins, sea-weed, plant gums), storage polysaccharides (starch-amylose and amylopectin), andlignin (plant cell walls) (Aguilera and Stanley, 1999).

C. Phase TransitionsPhase transitions, shown in the familiar state diagrams of biopolymers and

other food components, have a profound effect on the transport properties offoods, especially mass transport. Extensive data on the major phase transitions offoods, including freezing, glass transitions, gelatinization and crystallization, arepresented by Rahman (1995).

Food Structure and Transport Properties 31

Freezing of food materials increases substantially the thermal transportproperties (thermal conductivity and diffusivity), as shown in Chapter 8. Glasstransition, i.e. change from the glassy to the rubbery state, increases sharply themass diffusivity of water and other solutes (see Chapters 5, 7). Gelatinization ofstarch decreases moisture (solute) diffusivity, but it increases thermal conductivity(see Chapters 5, 8).

Collapse of the food structures reduces the mass transport properties offreeze-dried and other porous food materials. Collapse temperature is related to theglass transition temperature (Karathanos et al., 1996a).

Glass transition of food biopolymers and other components has receivedspecial consideration, because of its importance to the mechanical, transport, tex-ture and quality properties of several food materials and products (Roos, 1992;Roos, 1995; Rao and Hartel, 1998).

Due to their heterogeneous structure, solid foods show a diffuse but notsharp glass temperature change. The phase changes in food materials are deter-mined usually by differential scanning colorimetry (DSC). The glass transitiontemperature Tg increases considerably as the moisture content is reduced. The Tgof biopolymers is higher than the oligomers.

The effect of temperature on the transport properties (viscosity, diffusivity)below the glass transition temperature Tg and above (Tg+100K) is described by thefamiliar Arrhenius equation, while in the range of Tg to (Tg+100K) the Williams-Landel-Ferry (WLF) equation (see Chapter 7) gives a better representation.

D. Colloid and Surface ChemistryColloid and surface phenomena are important in the structure of most food

materials, especially liquid foods. A detailed analysis of the physicochemicalstructure of milk, the most important fluid food, is presented by Aguilera andStanley (1999).

Surface phenomena, like emulsions, foams, wetting and adhesion, affect thetransport properties, especially viscosity and mass diffusivity of foods. Therheological properties of liquid foods are particularly effected by the colloidalstructure of the food components (see Chapter 4). The apparent viscosity of mostcomplex (non-Newtonian) foods decreases considerably, as the shear rate is in-creased, due to changes in the flow patterns of the components in the suspension.

Viscoelastic (both viscous and elastic) phenomena in fluid foods are causedby intramolecular forces during deformation of the material. They are studied byrelaxation/creep experiments or by dynamic rheological tests.

32 Chapter 3

III. FOOD MICROSTRUCTURE AND TRANSPORT PROPERTIES

Food microstructure is concerned with the structure of food materials at themicroscopic level, and its relation to the processing, storage and quality of foodproducts. Developed initially for the evaluation of food texture and food quality,food microstructure can be applied to the transport properties of food materials,i.e. viscosity, thermal conductivity / diffusivity and mass diffusivity.

A. Examination of Food MicrostructureThe microstructure of foods can be measured and evaluated by the follow-

ing principal techniques (Aguilera and Stanley, 1999; Blonk, 2000):

1. Light microscopy (magnification x 20-500), which includes the compound,the polarizing, the fluorescent, the hot-stage, the computer-assisted and thecon focal laser scanning microscopes.

2. Transmission electron microscopy (TEM) with magnification (x200-500,000), which includes the scanning transmission electron microscope.

3. Scanning electron microscope (SEM) with magnification (x 20-200,000),which is considered as the best instrument for food microscopical studies andgives the best pictures of the materials.

In addition, the following techniques may be used for special stud-ies/examinations: a) scanning probe microscopy; b) X-ray microscopy; c) lightscattering; d) magnetic resonance imaging (MRI); and e) spectroscopy.

Image analysis relies on computer technology to recognize, differentiate andquantify images. It involves video cameras, scanners and data processing software.Image analysis is applied to the measurement of particle size and shape (Alien,1997) and in the control of food processing operations.

Fractal analysis is used to measure the irregularity of particle surfaces,which are of importance to food properties.

B. Food Cells and TissuesBoth plant and animal tissues consist of microscopic cells, which character-

ize each material. The cells contain several components, which are essential inliving organisms, such as water, starch, sugars, proteins, lipids and salts. A sche-matic diagram of a plant parenchyma cell is shown in Figure 3.1.


PLD

CM

CW

ML

Figure 3.1 Diagram of a parenchyma plant cell. CW, cell walls; CM, cell membrane(plasmalemma); V, vacuole; N, nucleus; ML, middle lamella; PLD, plasmodesmata; IS,intercellular space; TN, tonoplast; P, protein particles; L, lipid particles.

The microscopic plant cell (size of 2-10 um) consists of the cell walls whichcontain the cell components in a membrane (plasmalemma) enclosure. The cellcontains protein, starch and lipid particles within the cytoplasm. The vacuole, sur-rounded by the tonoplast, contains water, soluble sugars and salts (Aguilera andStanley, 1999). The vacuole is responsible for the osmotic pressure and the turgorof the cell.

The cell walls contain the middle lamella and they have small channels(plasmodesmata), that allow the flow of cytoplasmic material and water/solutes inand out of the cell. Intercellular spaces may contain water solution or air. The cellwalls contain mainly cellulose, hemicelluloses, pectins and glycoproteins. Planttissues consist of storage or parenchyma cells, phloem for transporting organicmaterials, xylem for transporting water and protective tissue.

The plant cell walls and membranes are of particular importance to foodprocessing and food quality. The turgor (hydrostatic pressure) is lost during dehy-dration, heating or freezing. The cell wall middle lamella complex is related tofood texture.

34 Chapter 3

Animal cells consist of fibrous structures (myofibrils), which have specialmechanical properties, characteristic of each animal material.

C. Microstructure and Food ProcessingFood structure is preserved in some processes, while in other it is destroyed

in order to produce useful processed products, as in refining starch, sugars, oilseeds, grain and milk. Changes in structure occur in freezing, milling (size reduc-tion), crystallization and emulsification. Restructuring of food materials is used inextraction, spinning, margarine and ice cream production.

Gels are solidlike structures that contain large amounts of water. They areproduced by different mechanisms using gelling substances, such as starch, pectin,gelatin, alginates and various plant gums. The Theological and transport propertiesof gels are important in the various food processing operations and in food quality.They are also used for investigations of structure-property relationships.

D. Microstructure and Mass Transfer

/. Solvent ExtractionSolvent/solid extraction or leaching of food components, such as sugars, lip-

ids and flavor compounds is used in various food processing operations. It can beanalyzed and designed by the methods developed in chemical process engineering,i.e. equilibrium-stage or continuous separations (Perry and Green, 1997).

The mass diffusivity of solutes in solid substrates Ds is smaller than the dif-fusivity in the liquid solvent DL due to the complex structure of the material, ac-cording to the empirical equation (Aguilera and Stanley, 1999):

Ds = Fa,DL (3-1)

The correction factor Fm varies in the range (0.1-0.9), with the higher valuesobtained when the cell membranes are destroyed, as by heating in the extraction ofsucrose from sugar beets by hot water. Fm becomes very low, approaching zero, inthe extraction of high molecular weight components, like proteins, from plant tis-sues. Extraction is improved by pretreatment of the solid material, e.g. by sizereduction, or by flaking, which reduces the diffusion path for both solvent andextracted component. In some applications, extraction can be improved by enzy-matic treatment of the plant tissues, which break down, e.g. the pectins of the cellwalls.


Values of the diffusivity of food components in various solvents are givenby Schwartzberg (1987). Typical DL for sucrose in water is 5xlO'10 m2/s and forcaffeine in water is IxlO"10 rrr/s (see Table 7.5).

2. Food DehydrationThe moisture (water) in heterogeneous solid foods may not be in equilib-

rium at the microstructural level, although macroscopically the system appears tobe in equilibrium. Thus, moisture may be transported between the heterogeneouscomponents of the food system. Thermodynamic analysis of transport processes atthe cellular level requires transport and equilibrium properties of the cell compo-nents (Rotstein, 1987).

Microstructural changes during the drying of food materials include loss ofcellular structure, pore formation and shrinkage of the product. These changesaffect strongly the transport properties, especially moisture diffusivity (see Chap-ter 5).

Heterogeneous structure may affect moisture diffusivity, like the reductionof the drying rate by the skin of grapes (see Chapter 5). Food microstructure isrelated to the retention of volatile aroma components during the drying of foodmaterials (see Chapter 7).

Microstructure plays an important role in the osmotic dehydration of foods,especially that of fruits and vegetables: The water is transported from the foodcells to the osmoactive solution (sugar or salt) and the osmoactive agent is infusedinto the cellular food (Lewicki and Lenart, 1995). A model of the osmotic dehy-dration process in cellular foods is presented by Yao and Le Maguer (1996).

Mass transfer in plant tissues during osmotic dehydration was analyzed bySpiess and Behsinlian (1998); the plant tissue is considered to consist of a solidmatrix, intercellular space, extracellular space and occluded gas. Three pathwaysof mass transfer may take place at the microstructural level: a) apoplasmatic trans-port (outside the cell membrane); b) symplasmatic transport through small chan-nels (plasmodesmata) between neighboring cells; and c) transmembrane transport(between the cell and the intercellular space). The function of living plant cells isexamined in plant physiology.

3. Microstructure and FryingFrying of foods can be considered as a simultaneous heat and mass transfer

process in which water is lost and oil is absorbed by the food product, e.g. potatopieces. Cells in the interior of the fried product can be intact, while surface cellsare dehydrated and shrunk. Oil uptake during frying is by a complex mechanism,different from the molecular (Fickian) diffusion, possibly by a capillary or hydro-dynamic flow (Aguilera and Stanley, 1999; Aguilera, 2000).

36 Chapter 3

IV. FOOD MACROSTRUCTURE AND TRANSPORT PROPERTIES

The transport properties of solid and semisolid food materials are related totheir macroscopic properties, such as density, porosity, particle size and shape.The design, operation and control of food processing operations is based on theseproperties, which can be measured by simple instruments and techniques.

The structure of solid and semisolid foods has been investigated more in re-lation to food structure and food quality than to transport properties. The structureof fluid foods is usually related to their rheological and viscometric properties, asdiscussed in Chapter 4.

Food macrostructure has received special attention in relation to the dehy-dration of foods, since significant changes take place during moisture transport insolid and semisolid food materials.

Model food materials, based on food biopolymers, such as starch, and fruitsand vegetables have been used as experimental materials, since their propertiescan be related empirically to their structure at the macroscopic, microscopic andmolecular levels.

Quantitative parameters of physical meaning, i.e. density, porosity andshrinkage, based on three phases (solids, water and air) can be estimated for thecharacterization of structural changes of foods during processing and storage.

Two classes of food materials are examined separately: a) continuous solids,in which shrinkage and porosity develop when water is removed, and b) particu-late or granular materials, such as starch granules, in which porosity is a dependentvariable that can be controlled, e.g. by compression.

A. DefinitionsAssuming moist material to consist of dry solids, water, and air, the follow-

ing definitions can be considered:

m,= ms+mw (3-2)

where m,, ms, and mw are the total mass and the masses of dry solids and waterrespectively (kg), while the mass of air is neglected.

The total volume of the sample is considered as:

V,= Vs+Vw+Va (3-3)

where Vs, Vw and Va, are the volumes of dry solids, water and air pores, respec-tively (m3). The volume of air is referred to the internal pores only.


The apparent density of the food material pb is defined as:

pb = m,/Vt (3-4)

and the true density pp as:

PP=m,/Vp (3-5)

where Vp = Vs+Vw is the true (particle) volume, which is the total volume of thesample excluding air pores. Apparent and true density are analogous to the bulkand particle density of granular materials, respectively. The actual densities of drysolids PJ and enclosed water pw can also be defined as:

P, = m,/V, (3-6)

pw = mw/Vw (3-7)

The specific volume of the sample u is defined as the total volume per unit mass ofdry solids (m3/kg db):

v=V,/ms (3-8)

The material moisture contention a dry basis (kg water/kg db) is:

X=mw/ms (3-9)

The volume-shrinkage coefficient ft can be defined by the following equation,which represents the proportion of initial specific volume that shrinks as water isremoved:

(3.10)A,

where Xi is the initial moisture content of the moist food material, v is the specificvolume at material moisture content X, and o,- is the specific volume at X=Xi. Theshrinkage coefficient ft varies between 0 (no shrinkage) and 1 (full shrinkage).(See Figure 3. 2.)

Assuming that no volume interaction occurs between the water and the sol-ids, combining Eqs. (3-5), (3-6), (3-7) and (3-9) results in:

38 Chapter 3

Pp= lJrX (3-11)

A A..

Equation (3-11) shows the dependence of moisture content on true (particle) den-sity.

Combining Eqs. (3-4), (3-7), (3-9) and (3-10) results in:

'iW P-12)A, A,.

where pbi is the apparent density &iX=Xi, the initial moisture content.When the zero moisture content is considered as initial moisture content

(A)=0), then Eqs. (3-10) and (3-12) are transformed into Eqs. (3-13) and (3-14),respectively:

o = uo+/3— (3-13)A

where L>O, and pbo are the specific volume and the apparent density at X = 0, re-spectively.

Moreover, in fried products, when oil is considered as one more phase, Eqs.(3-13) and (3-14) are further transformed to Eqs. (3-15) and (3-16), respectively:

( X Y\o = v a+p\ — + — (3-15)I A PL)

X Y— + —A» A PL.

where pL is the oil density and 7 the oil content

Y=mL/ms (3-17)


where mL is the mass of the oil in the sample of ms dry solids.For fried products Eq. (3-11) is also transformed to Eq. (3-18):

+ X +1 X Y

_____ j ___ I ___

P, P, Pi

(3-18)

Volume of water which:

disappears (shrinkage)= p (Xi-X) / pw

remains as water= X/pw

remains as air (porosity)= (l-p)(Xi-X)/pw

Initial Final(Xi) (X)

Figure 3.2 Schematic representation of shrinkage and porosity development, aspart of water is removed.

40 Chapter 3

B. Food Macrostructure and Transport Properties

The food macrostructure, in calculating the effective transport properties offood materials, is taken into account, using the so-called structural models, someof which are presented for thermal conductivity in Chapter 8 (Table 8.3). Similarstructural models for moisture diffusivity of starch materials have been proposedby Vagenas and Karathanos (1991), but their application to food materials remainsto be validated.

These models require the volume fraction of the food phases (solids, water,air, oil, etc). Thus, the shrinkage and porosity models, described earlier in thischapter, must be combined with the transport properties models.

Two examples for continuous (Maroulis et al., 2001; Krokida et al., 200la)and granular foods (Krokida et al., 200Ib) foods follow.

1. Continuous SolidsTable 3.1 summarizes the proposed model that combines the thermal con-

ductivity structural models with the density structural models. The correspondinginformation flow diagram is presented in Figure 3.3.

2. Granular SolidsTable 3.2 summarizes the proposed model, which combines the thermal

conductivity structural models with the density structural models. The correspond-ing informational flow diagram is presented in Figure 3.4.


Table 3.1 Effective Thermal Conductivity Generic Model for Continuous Mate-rials

Thermal Conductivity Structural Model

Density Structural Model

n

(2)

Volume Fraction of the Food Phases

/ ? ,= ! -— (4)

* =^ (5)

(6)

A, P.

42 Chapter 3

O, al, a2, a3

Densityof Pure

Substances

ThermalConductivity

of PureSubstances

ps, pw

pai, Xi,

Shrinkageand

PorosityModel

Eqs. (7) and (8)

pa, pt

VolumeFractionof Food

ComponentsEqs. (4), (5)

and (6)

Xs, Xw, Xa

bo, bl, b2, b3

es, e\v, ea

ThermalConductivity

StructuralModel

Eqs. (1), (2)and (3)

JXeff

Figure 3.3 Informational flow diagram for the model of Table 3.1. (Equation numbersrefer to those in Table 3.1.)


Table 3.2 Effective Thermal Conductivity Generic Model for Granular Materials

Thermal Conductivity Structural Model for Granular Material

*«=l-f f < ! >-L J-

(2)

(3)

Thermal Conductivity Structural Model for Granules (Particles)

/y f

(5)

Volume Fractions

*0=1-A (7)^

Particle Density Structural Model

P = l + X (9)Pp 1 X ^ '___ I ___

P. P.

44

aO, al, a2. a3

Densityof Pure

Substances

pb

ThermalConductivity

of PureSubstances

bo, bl, b2, b3

ps, pw

Xs, Xw

Xa

Chapter 3

f fp

Particle DensityStructural Model

Eq. (9)

PP

VolumeFractions

Eqs. (7) and (8)

Thermal ConductivityStructural Model

for Granules (Particles)Eqs. (4), (5)

and (6)

Xp1 e1

Thermal ConductivityStructural Model

forGranular MaterialEqs. (1), (2)

and (3)

Xeffl

Figure 3.4 Informational flow diagram for the model of Table 3.2. (Equation numbersrefer to those in Table 3.2.)


C. Determination of Food Macrostructure

The determination of food macrostructure is based on the measurement ofmass and volume and the estimation of the various densities and porosity, asshown on the informational flow diagram of the Figure 3.5 (Krokida et al., 1997).The corresponding methods are described by Rahman (1995) and summarized inTable 3.3.

Mass of Wet Samplem

(error 1%)

True VolumeVp

(error 1%)

Total VolumeVt

True Densityp p =m, /Vp(error 2%)

Apparent Densityp b = m t / V t(error 3%)

(error 2%)

Mass of Dried Samplems

(error 1%)

Specific Volumeu = Vt / ms

(error 3%)

Figure 3.5 Experimental data evaluation flow diagram for densities and porosity.

46 Chapter 3

Table 3.3 Volume Measurement Techniques

1. Dimension measurement2. Buoyant force determination3. Volume displacement method

a. Liquid displacement methodb. Gas pycnometer methodc. Solid displacement method

D. Macrostructure of Model Foods

Model foods are useful experimental materials for studying the effect ofphysical (macro) structure on the transport properties of foods. They consist of abiopolymer matrix, containing water and typical food components, such as carbo-hydrates, proteins and lipids assembled in the form of a hydrated solid or a solid-like gel. Starch materials, particularly linear amylose and branched amylopectin,have been used in many forms for studies of thermal and mass diffusivity. Thephysical phenomena of food dehydration have been investigated using variousstarch materials (Saravacos, 1998).

The density of granular starch changes nonlinearly with the moisture con-tent, according to the empirical equation (Marousis and Saravacos, 1990):

pf = 1442 + 837X-3646X2 + 448 LY3-1850^4 (3-19)

The granular or particle density pp is expressed in (kg/m3) and the moisturecontent A'in (kg/kg db).

Figure 3.6 schematically shows the change of the particle density of granu-lar starch as a function of moisture content. A maximum ofpp is observed near X= 0.15, meaning that at this low moisture content, the water is adsorbed stronglyon the biopolymer. At higher moistures, water reduces the particle density byswelling the starch granules.

The bulk porosity s of hydrated granular starch increases significantly as themoisture is reduced during air-drying (Figure 3.7). A smaller change of porosity isobserved on the porosity of gelatinized starch during the drying process. Thesesignificant changes in porosity, estimated by measuring the bulk and particle den-sities of the material, are related to the changes of moisture diffusivity D duringthe various drying and rehydration processes (see Chapter 5).


1.60

1.000.00 0.20 0.40 0.60 0.80 1 .00

X (kg/kg db)

Figure 3.6 Particle density pp of corn starch as a function of moisture content X.

0.60

0.40

0.20

0.000.00 0.20 0.40 0.60 0.80 1.00

X (kg/kg db)

Figure 3.7 Bulk porosity s of air-dried granular (GR) and gelatinized (GEL) cornstarch as a function of moisture content^

48 Chapter 3

Gelatinization of starch increases significantly the thermal conductivity ofthe starchy materials, evidently due to the changes in the macromolecular and mi-croscopic structure (see Chapter 8). Changes in porosity of granular starch struc-tures by the incorporation of sugars, proteins, lipids, and inert particles are re-flected in significant changes of the moisture diffusivity. Significant reduction ofthe porosity of granular starch is obtained by mechanical compression (see Chap-ter 5).

Figure 3.8 shows schematically the macrostructure of dried granular and ge-latinized starch materials. The spherical granular starch materials developed radialchannels through which water was transported by hydrodynamic flow during dry-ing. The gelatinized starch suffered more shrinkage during drying with irregularcracks in the dried matrix. These changes correspond to changes of moisture diffu-sivity during the drying processes (Marousis and Saravacos, 1990).

Freeze-drying of model food gels affects strongly their macrostructure andthe transport properties. The development of macrostructure is the combined resultof freezing and drying by sublimation of the ice. Figure 3.9 schematically showsslabs of two different structures, developed by freeze-drying of model gels. TheCMC gel developed a fibrous structure, parallel to the flow, which significantlyincreased the moisture transport rate and thermal conductivity of the material (seeChapters 5 and 8). The freeze-dried starch gel had a uniform microporous struc-ture which had lower moisture and thermal diffusivity than the fibrous material.

The pore size distribution in granular, gelatinized and extruded starch mate-rials, measured by mercury porosimetry, shows that the majority of the pores(90%) are larger than 1 urn, and only 10% have smaller pores. Bulk porosity,measured by gas pycnometry, is the most important parameter characterizing thetransport properties (Karathanos and Saravacos, 1993). Structural models, devel-oped for thermal conductivity, can be adapted to model the effective moisture dif-fusivity in granular starch materials (Vagenas and Karathanos, 1991). The porestructure of extruded pasta has significant effect on the moisture sorption and dif-fusivity of pasta (Xiong et al., 1991).

Stress crack formulation in the air-drying of cylindrical samples of hydratedstarch is affected by the moisture gradient at the transfer interface (Liu et al.,1997). Crack formation, a problem in pasta drying, is related to the stresses devel-oped in the glassy state of the biopolymer, and it can be prevented by drying attemperatures higher than the glass transition temperature (Willis et al., 1999). De-velopments of cracks in grain kernels can be prevented by intermittent microwavedrying, due to the relaxation of temperature and moisture gradients during thetempering period (Zhang and Mujumdar, 1992). Tailor-made porous solid foodscan be prepared using a base of freeze-dried alginate gels (Rassis et al., 1997).


a. Granular b. Gelatinized

Figure 3.8 Macrostructure of air-dried spherical samples of corn starch (20 mm diame-ter).

O - OO Q O

a. CMC b. Starch

Figure 3.9 Macrostructure of freeze-dried slabs of carboxy methyl cellulose (CMC) andstarch gels (20 mm thickness).

50 Chapter 3

E. Macrostructure of Fruit and Vegetable MaterialsThe effect of drying method on bulk density, particle density, specific vol-

ume and porosity of banana, apple, carrot and potato at various moisture contentsis presented in Figures 3.6-3.9, using a large set of experimental measurements(Krokida and Maroulis 1997). Samples were dehydrated with five different dryingmethods: conventional, vacuum, microwave, freeze- and osmotic drying. A simplemathematical model, presented in Table 3.4, was used in order to correlate theabove properties with the material moisture content. Four parameters with physi-cal meaning were incorporated in the model: the enclosed water density pw, thedry solids density ps, the dry solids bulk density pbo and the volume shrinkage co-efficient (3. The effect of drying method on the examined properties was taken intoaccount through its effect on the corresponding parameters (Table 3.5). Only drysolid bulk density was dependent on both material and drying method. Freeze-dried materials developed the highest porosity, whereas the lowest one was ob-tained using conventional air drying.

The above structural properties for the investigated materials were also ex-amined during rehydration of dehydrated products, using the drying methods. Thesame dehydrated products did not recover their structural properties after rehydra-tion, due to structural damage that occurred during drying and the hysteresis phe-nomenon, which took place during rehydration. Porosity of the rehydrated prod-ucts was higher during rehydration than during dehydration. A structural model ofTable 3.4 was also used to describe the structural properties, and of the four pa-rameters that were incorporated (also presented in Table 3.5), only the shrinkagecoefficient, which represents volume expansion, changed on rehydration.

Apparent density, true density, specific volume and porosity were investi-gated during deep fat frying of french fries (Krokida et al., 2000). The effect offrying conditions (oil temperature, sample thickness and oil type) on the aboveproperties is presented in Figures 3.10-3.12. Moisture and oil content during deepfat frying and consequently all the examined properties are affected by frying con-ditions. The results showed that the porosity of french fries increases with increas-ing oil temperature and sample thickness, and it is higher for products fried withhydrogenated oil.

The pore size distribution of fruit and vegetable materials shows three peaksin the ranges of 20 um, 1 um and 0.2-0.04 urn. The pore size of air-dried foodmaterials is much smaller than the pore size of freeze-dried products, due to thecollapse of structure during dehydration (Karathanos et al., 1996b).


2.0 -

APPLE

2.0 TBANANA

AX

Dehydrationconvective dryingvacuum dryingmicrowave dryingfreeze dryingosmotic dehydrationcalculatedRehydrat ionconvective dryingvacuum dryingmicro\\ave dryingfreeze dryingosmotic dehydration

- calculated

0 3 6Moisture content (kg/kg db)


2.0 - 2.0 -

CARROT



Figuremethods

3.10 Variation of true density with material moisture content for various dryingduring dehydration and rehydration.

52 Chapter 3

2.0 -

0.0

iooD

Dehydconveivacuuimicro\freezeosmoti

Rehydconvetvacuuimicrovfreezeosmoti'

(ration;tive drying•n dryingvave dryingdrying

c dehydrationitedIration;tive dryingn dryingvave dryingdrying

c dehydration

0 3 6

Moisture content (kg/kg db) Moisture content (kg/kg db)

0.0

CARROT

0 3 6

Moisture content (kg/kg db)0 3 6

Moisture content (kg/kg db)

Figure 3.11 Variation of apparent density with material moisture content for variousdrying methods during dehydration and rehydration.


1.0 T BANANA •AX

Dehydrationconvective dryingvacuumdryingmicrowavedryingfreeze d ry ingosmotic d e h y d r a t i o n

. ca lcu la tedRehydrationconvective dry ingv a c u u m d r y i n gmicrowave dryingfreeze dry ingosmotic dehydration

-calculated

Moisture content (kg/ kg db) Moisture content (kg/ kg db)

1,0 T

CARROT

0 3 6

Moisture content (kg/kg db)0 3 6

Moisture content (kg/ kg db)

Figure 3.12 Variation of porosity with material moisture content for various drying meth-ods during dehydration and rehydration.

54 Chapter 3

15 •AX••

oAX0D

Dehydrationconvective dryingvacuum dryingmicrowave dryingfreeze dryingosmotic dehydration

Rehydrationconvective dryingvacuum dryingmicrowave dryingfreeze dryingosmotic dehydration

3 6Moisture content (kg/kg db)


15 -



Figure 3.13 Variation of specific volume with material moisture content for various dryingmethods during dehydration and rehydration.


Table 3.4 Mathematical Model for Structural Properties of Foods

Structural propertiespp True densitypb Apparent densitys Porosityv Specific volume

Factorsx Moisture content

Properties equations1 + X

Pp ~ J_ APs P.

i + xpb = • t

Pbo Pw

8 = 1-*-Ppu = —+ B' —

Pbo Pw

(kg/L)(kg/L)

(L/kg db)

(kg/kg db)

(1)

(2)

(3)

(4)

Parameterspw Enclosed water densityps Dry solid true densitypbo Dry solid apparent density/?' Shrinkage or expansion coefficient

Factors affecting parametersMaterialDrying method

(kg/L)(kg/L)(kg/L)

56 Chapter 3

Table 3.5 Parameter Estimation of the Structural Properties Model

Me'£•c

Reh

ydra

t.

DCB

'E>Q

Reh

ydra

t.

itc'E>O

u•B>,

-C

£

BJDC

'E-c

uJ>i

Material/Method

AppleConvective

VacuumMicrowave

FreezeOsmotic

ConvectiveVacuum

MicrowaveFreeze

OsmoticBanana

ConvectiveVacuum

MicrowaveFreeze

OsmoticConvective

VacuumMicrowave

FreezeOsmoticCarrot

ConvectiveVacuum

MicrowaveFreeze

ConvectiveVacuum

MicrowaveFreezePotato

ConvectiveVacuum

MicrowaveFreeze

ConvectiveVacuum

MicrowaveFreeze

Ps P» P

0.990.961.010.34

• - '- -g-1.311.300.811.22

1.040.901.050.431.04

' L9° L°2 1.071.101.070.651.07

1.020.990.940.30

1.75 1.02 12Q

1.051.200.22

1.031.030.810.29

1.60 1.02 10?

1.051.070.74

Pbo

0.560.390.560.120.730.560.390.560.120.73

1.810.631.790.261.331.810.631.790.261.33

1.600.920.530.141.600.920.530.14

1.501.290.440.181.501.290.440.18


1.5 n

0.9

0.6

5 10 15 20Time (min)

5 10 15 20Time (min)

5 10 15 20Time (min)

5 10 15 20Time (min)

Figure 3.14 Effect of oil temperature on structural properties of french fries.

(Q CO ^ 01

Poro

sity

o

oTr

ue d

ensi

ty (k

g/1)

<n CO

H 3

s I •a o 31.

Spec

ific

volu

me

(I/k

g)U

J

LK>

4^

App

aren

t den

sity

(kg/

1)o

o

O

<-*»

O

o -

' '

<-/!

d 3 re I5

"

5" x^<_ft K

)

\ 1 1

Lft 3 3

r • | <

1 1

I 1

r> • \ • 0 3 3

/ • (-A 3 3

O

5"

O 3- 9> T3 ff


1.5 -i

0.

Concentration ofhydrogenated oil (%)

A O• 50• 100

5 10 15Time (min)

20 5 10 15Time (min)

20

5 10 15 20Time (min)

0 5 10 15 20Time (min)

Figure 3.16 Effect of oil type on structural properties of french fries.

60 Chapter 3

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Vagenas, O.K., Karathanos, V.T. 1991. Prediction of Moisture Diffusivity inGranular Materials with Special Applications to Foods. Biotechnol Progr7:419-426.

Vieth, W.R. 1991. Diffusion In and Through Polymers. Munchen, Germany: Han-ser.

Willis, B., Okos, M., Campanella, O. 1999. Effect of Glass Transition on StressDevelopment During Drying of a Shrinking Food System. In: Proceedings ofthe 6th CoFE '99. Barbosa-Canovas, G.V., Lombardo, S., eds. New York:AIChE, pp. 496-501.

62 Chapter 3

Xiong, X., Narsimhan, G., Okos, M. O. 1991. Effect of Composition and PoreStructure on Binding Energy and Effective Diffusivity of Moisture in PorousFoods. J Food Eng 15:187-208.

Yao, Z., Le Maguer, M., 1996. Mathematical Modeling and Simulation of MassTransfer in Osmotic Dehydration Processes. 1. Conceptual and MathematicalModels. J Food Eng 29:349-360.

Zhang, D., Mujumdar, A.S. 1992. Deformation and Stress Analysis of PorousCapillary Bodies During Intermittent Volumetric Thermal Drying. DryingTechnol 10:421-443.

Rheological Properties of Fluid Foods

I. INTRODUCTION

The viscosity of fluid foods is an important transport property, which is use-ful in many applications of food science and technology, such as design of foodprocesses and processing equipment, quality evaluation and control of food prod-ucts, and understanding the structure of food materials. Due to the complexchemical and physical structure of foods, viscosity can not be predicted by theo-retical methods, such as molecular dynamics and semi-empirical models, appliedto pure fluids, and discussed in Chapter 2 of this book. Therefore, experimentalmeasurements and empirical models of viscosity are necessary for the characteri-zation of fluid foods.

Viscosity is part of the wider rheological properties of foods, which cover,in addition to fluids, the solid and semisolid food materials. Foods, in general, canbe classified as solids, gels, homogeneous liquids, suspensions in liquid, andemulsions (Rao, 1999). Fluid foods are heterogeneous materials, consisting ofdispersions of fibers, cells, protein particles, oil droplets and air bubbles in a con-tinuous phase, like an aqueous solution of sugars, or a vegetable oil (Aguilera andStanley, 1999).

Recent advances in the design and control of food processes, utilizing com-puter modeling and simulation, require extensive data on the physical and engi-neering properties of foods. Limited reliable data are available in the literature,particularly in the areas of rheological properties (viscosity) and mass diffusivityof food systems (Saravacos and Kostaropoulos, 1995, 1996; Saravacos, 2000).

Food rheology deals with all the phenomena of deformation and flow offood materials due to external forces. Viscometry deals with fluids, which arecharacterized by mechanical flow, upon the application of an external force.

63

64 Chapter 4

The viscometric properties of fluid foods are discussed here in analogy tothe two other basic transport properties, thermal conductivity and mass diffusivity.The rheological properties of solid and semisolid foods (elasticity and viscoelastic-ity) are discussed in specialized books, such as Bourne (1982) and Rao (1999).

Viscosity plays an important role in liquid food texture and in texture-tasteinteractions (Kokini, 1987). The tasting reaction is controlled by the diffusion ofthe flavor components through the viscous food layer in contact with the tongue.

Fluid viscosity rj is defined by the basic transport Eq. (2-2), which is equiva-lent to the Newton Eq. (4-1) of shear flow. In most food applications, TJ is syn-onymous to the shear viscosity in the -direction, as shown in Figure 4.1.

T = riy (4-1)

where T = F/A (applied force / surface area) is the shear stress in (Pa) andy = (du/dy), change of velocity (Aux) in the y-direction, is the shear rate in (1/s).

This book deals mainly with shear viscosity at steady state. However, thereare some other types of viscosity, for example extensional viscosity, which is use-ful in specialized engineering applications. Extensional flows occur when the flowgeometry changes abruptly, like in orifice flow, spinning of fibers, extrusion, andimpingement (Giesekus, 1983; Padmanabhan, 1995; Rao, 1999). Such flows areimportant in polymer processing. In the simple uniaxial extensional flow, a cylin-drical body is stretched in one direction, while contracting in the other two. In thebiaxial extensional flow, the reverse process takes place, i.e. the material isstretched in two directions, while contracting in the other. Finally, in the planarextensional flow, stretching is in one direction, constant in the second, and con-traction in the third.

Figure 4.1 Shear viscosity in the x-direction.

Rheological Properties of Fluid Foods 65

In dynamic rheometry, applied to viscoelastic materials and gels, the shearstress-strain relationship is obtained by periodic deformation of the material (Raoand Steffe, 1992; Urbicain and Lozano, 1997).

In polymer science, the intrinsic viscosity [ r/J expresses the hydrodynamicvolume of the polymer molecule, which is related to the molecular weight and thedimensions of the molecule. The intrinsic viscosity in dilute polymer solutions isdefined as the zero-concentration limit of the ratio (rjS[/C), where T]sp

=[('n-ris)/riJis the specific viscosity, r\ and rjs are the viscosity of the solution and the solventrespectively, and C is the concentration (Rao, 1999).

The intrinsic viscosity [ t] /of dilute food biopolymer solutions (e.g. gums)has been related to experimental viscosity measurements through empirical mod-els, like the Huggins equation (Tanglertspaibul and Rao, 1987; Yoo et al, 1994):

(iv'Q-lvJ + krfrifC (4-2)

The Huggins constant kt is related to the polymer-polymer interaction and ittakes values from 0.3 to 1.0. The low kj values characterize the good solvents,while the high values indicate associations between the macromolecules. Heattreatment of aqueous gum solutions decreases the intrinsic viscosity, presumablydue to hydrolysis or breakdown of the biopolymer (Rao, 1995).

Real solids are considered as elastic materials, which obey Hooke's law, i.e.a linear relationship between shear stress rand strain /.

T =EY, or T =GY (4-3The Young's modulus of elasticity E is used when the applied strain y is the

elongation or compression, y = Al /1. The shear modulus G is used when / is theshear strain.

Most solid and semisolid foods are not elastic, but they behave as either vis-coelastic or viscoplastic. The viscoelasticity of the materials is studied by stressrelaxation measurements, i.e. shear stress at constant strain versus time (Rao,1999). The viscoelastic behavior is characterized by Newton's law (Eq. 4-1),Hooke's law (Eq. 4-3), and Newton's second law (F = m a, where m is the massand a is the acceleration).

Most solid and semisolid foods are considered linear viscoelastic, which al-lows the adding up of the three elements, i.e. the viscous, elastic, and inertial ef-fects. Neglecting the inertial component, the viscous (dashpot) and the elastic(spring) effects are usually combined into two common models, i.e. the Newton(series model), and the Kelvin-Voigt (parallel model).

66 Chapter 4

II. RHEOLOGICAL MODELS OF FLUID FOODS

The rheological behavior of fluid foods is determined by measurements ofshear stress versus shear rate, and representation of the experimental data by vis-cometric diagrams and empirical equations, as a function of temperature and/orconcentration. Molecular dynamics can not predict fluid viscosity, but it can behelpful in understanding the flow mechanism of complex fluid foods. Physicalstructure plays a decisive role in determining the fluid viscosity.

A. Structure and Fluid ViscositySimple liquid food materials, like aqueous sugar solutions, clarified juices,

and vegetable oils are Newtonian fluids, i.e. the shear stress is linearly propor-tional to the shear rate, according to Newton's law (Eq. 4-1). Incorporation ofpolymer molecules and micelles, solid particles, droplets, and gas bubbles intoNewtonian fluids changes considerably the rheology of fluid foods, evidenced bythe non-linear relationship of shear stress/shear rate.

Model fluid suspensions are useful in understanding the flow behavior of thecomplex fluid food systems. The viscosity of fluid suspensions reflects the com-plex hydrodynamic flow of the particle / solvent and particle / particle interactions.The particle size, shape and concentration affect strongly the viscometric proper-ties of fluid foods.

In a simple suspension of spherical particles, the relative viscosity of thesuspension to the viscosity of the continuous phase Tjr is a linear function of thevolume fraction of the particles (/), according to the Einstein equation (Giesekus,1983):

rjr = 1 + 2.5 <f) (4-4)

The Einstein equation holds for dilute concentrations of spherical particles,which do not react with the solvent and with each other. Particle-solvent and parti-cle-particle interactions are reflected by a non-linear relationship of (rjr, (f>). Theshape of the particles affects strongly the behavior of suspensions Figure 4.2.Thus, the deviation from linearity (Eq. 4-2) increases when the particle shapechanges from spherical to plate and rods. Still, stronger effects are observed withfibers of increasing aspect (L/D, length / diameter) ratios.

Attracting forces, acting on the particle surfaces, tend to create aggregates,which significantly increase the viscosity of the suspension. Increasing the shearrate, the aggregates are broken down, resulting in shear-thinning of the suspension.The suspended non-spherical particles give higher shear viscosity, due to the diffi-culty of rotating (flowing) in the suspension. The aspect (L/D) ratio of rodlike par-


tides (e.g. fibers) is related to the build-up of transient network structures. Shearthinning may be found at low concentrations of rodlike particles (e.g. fibers).

Model suspensions of coarse food particles in aqueous CMC solution ex-hibit Theological properties similar to inorganic models, i.e. their deviation fromNewtonian behavior increases when the particle shape becomes elongated, andwhen the particle concentration is increased (Pordesimo et al., 1994).

The non-linearity of suspensions of spherical particles at high concentrationscan be described by empirical equations, such as the Frankel-Acrivos and Krieger-Dougherty models, in which the relative viscosity (t|r) is related to the volumefraction ratio ($/$„), where </>m is a shear-dependent maximum volume fraction(Rao, 1999).

Food particles are usually non-spherical, and they resemble to rough crys-tals. The empirical Kitano et al. equation (1981) can be applied:

TJr=[l-(j/A)]

where A is a constant, A = 0.68 for spheres, and A = 0.44 for crystals.

(4-5)

Figure 4.2 Relative viscosity r]r of suspensions of glass particles as a function of thevolume fraction </> in water. (1) Einstein's equation, (2) spheres, (3) plates, (4) rods.(Adapted from Giesekus, 1983.)

68 Chapter 4

Plots of Eq. (4-5) resemble the plots of Figure 4.2. The Kitano et al. modelwas applied to reconstituted tomato puree of narrow particle size distribution withaverage diameters 0.71 and 0.34 mm, and adjusting parameters A = 0.54 and A =0.44, respectively (Yoo and Rao, 1994).

Fluid foods are complex suspensions of hydrophilic particles that interactphysically and chemically with the solvent (water), resulting in considerabledeviations from Eq. (4-2). The particle size is affected by the various processingoperations and storage conditions. Thus, the size of starch granules increasesconsiderably during heating, before gelatinization, resulting in increased relativeviscosity of aqueous suspensions (Rao, 1999). Changes in particle size of tomatoand other fruit and vegetable products, during pulping and screening operations,will strongly affect the product viscosity. Physicochemical interactions in fluidfoods, such as adsorption, emulsification, polymer conformation, de-poly-merization, crystallization, and melting have significant effects on viscosity.

B. Non-Newtonian FluidsFluid foods, containing dissolved macromolecules and suspended particles,

deviate considerably from Newtonian behavior (Eq. 4-1). Various empirical mod-els have been used to account for the observed non-linear relationship of shearstress (T)- shear rate (y) of the non-Newtonian fluids (Holdsworth, 1993; Rao,1987a, 1987b).

1. Time-Independent ViscosityThe usual non-Newtonian models apply to time-independent fluid foods,

for which the shear stress-shear rate relationship does not change with the time ofshearing.

The Bingham plastic model (Eq. 4-6) applies to Newtonian fluids that willflow only if a fixed yield stress TO is exceeded:

T = TO + TIY (4-6)Many non-Newtonian fluid foods are represented by the power-law or

Ostwald-de Waele model (Eq. 4-7):

T = Ky" (4-7)

In the power-law model, two rheological constants (K, n) are required tocharacterize the flow behavior, K the flow consistency coefficient or index withunits (Pa sn), and n the flow behavior index (dimensionless). The K value corre-sponds to the viscosity of Newtonian fluids.

The Herschel-Bulkley model is derived from the power-law model by add-ing a yield stress term r0:


T = Tn (4-8)

Most non-Newtonian foods are pseudoplastic materials (n<l), while veryfew are dilatant (n>Y). The pseudoplastic fluids are also known as shear-thinningfluids, since their apparent viscosity decreases as the shear rate is increased.

The power law and Herschel-Bulkley models have been used widely, andextensive rheological data for non-Newtonian fluid foods have been published inthe literature (Prentice and Huber, 1983; Okos, 1986; Kokini, 1992; Urbicain andLozano 1997; Rao, 1999). Figure 4.3 shows typical rheological diagrams, r versusY, for various non-Newtonian fluids.

The apparent viscosity of non-Newtonian fluids is an important property,which can be used in several engineering and technological applications in theplace of Newtonian viscosity. The apparent viscosity rja of power-law andHerschey-Bulkley fluids is defined by the equation

.n-l (4-9)

In pseudoplastic (shear-thinning fluids), (n-l) < 0, and therefore the apparentviscosity decreases as the shear rate y is increased.

Shear Rate

Figure 4.3 Shear stress C^-shear rate (y) diagrams for Newtonian (1), Bingham(2), pseudoplastic (3), and dilatant (4) fluids.

70 Chapter 4

Specialized rheological models have been applied to certain fluid foods,such as the Casson model for chocolate:

ras = r^+Kcy0'* (4-10)

The Casson plastic viscosity rjCa is calculated as the square of the Cassoncoefficient (Rao, 1999):

Ka=(KJ2 (4-11)

The Mizrahi-Berk model (Eq. 4-12) is a modification of the Casson equa-tion, and it has been applied to concentrated orange juices (Mizrahi and Berk,1972):

T™ = *„«*+&?" (4-12)

The rheological constants (Kc, n) are characteristic for a fluid at a giventemperature/concentration.

2. Time-Dependent ViscosityMany non-Newtonian fluid foods of complex structure exhibit time-

dependent rheology, i.e. their apparent viscosity at a given shear rate changes sig-nificantly with time of shearing. The most common time-dependent materials arethe thixotropic fluids, for which the apparent viscosity decreases with the time ofshearing. The rheopectic fluids exhibit the opposite behavior, i.e. the apparent vis-cosity increases with the time of shearing. The time-dependent rheological behav-ior is detected usually by the rheological diagram (T versus y), which forms acharacteristic loop, when ascending and descending values of stress rare plottedagainst the corresponding shear rate y. In thixotropic fluids, the descending linefalls below the ascending, while the opposite behavior is observed in the rheopec-tic fluids (Figure 4.4).

Empirical models have been proposed to descibe the thixotropy of fluidfoods, in which the shear stress is a function of time, with the initial and equilib-rium stresses as constants.


VI!_«CJ

JSV)

Shear Rate

Figure 4.4 Shear stress (r)-Shear rate (y) diagrams (loops) for time-dependent fluids:(1) thixotropic, (2) rheopectic.

C. Effect of Temperature and Concentration

In pure liquids, the effect of temperature on viscosity follows the Arrheniusequation (2-31), which can be derived from the theory of rate processes. The sameequation has been applied to the viscosity 77 of Newtonian fluid foods, and to theconsistency coefficient K or the apparent viscosity rjaof power-law (non-Newtonian) food materials (Rao, 1999):

= K0exp(Ea/RT) (4-13)

where K0 is a frequency factor (Pa s"), Ea is the energy of activation for viscousflow (kJ/mol), T is the temperature (K), and R = 8.314 kJ /kmol K is the gas con-stant.

72 Chapter 4

In Newtonian fluid foods, the energy of activation has been found to in-crease from 14.4 kJ/mol (water) to more than 60 kJ/mol (concentrated clear juicesand sugar solutions).

Temperature has a major effect on the consistency coefficient K and the ap-parent viscosity i]a of the non-Newtonian fluid foods, analogous to the effect onNewtonian viscosity rj. The flow behavior index n is affected only slightly bytemperature (a small increase at high temperatures). The energy of activation forflow in non-Newtonian fluids is significantly lower than the corresponding valuefor Newtonian fluids of the same solids concentration (Saravacos, 1970). In sus-pensions of fluid foods of high non-soluble solids concentration, like fruit or vege-table pulps, Ea may be lower than the activation energy for viscous flow of water(14.4kJ/mol).

The observed strong effect of temperature on the viscosity of viscous New-tonian fluid foods (concentrated clear juices, edible oils) is similar to the effect onthe viscosity of nonpolar viscous liquids, such as mineral oils, as is described bythe Lewis-Squires Eq. (2-33).

Concentration of soluble solids (°Brix) and insoluble solids (e.g. pulp) has astrong non-linear effect on the viscosity of Newtonian fluid foods, the consistencycoefficient K, and the apparent viscosity of non-Newtonian foods. Two similarexponential models, one for °Brix and a second for % pulp, were proposed by Vi-tali and Rao (1984a, 1984b) for concentrated orange juice, of the general form:

K = K0exp(BC) (4-14)

where K0 is a frequency factor, C is the concentration and B is a constant.The combined effect of temperature and solids concentration can be ex-

pressed by combining Eqs. (4-13) and (4-14):

K = K0 exp [ (Ea / RT) +BC] (4-15)

The additive model assumes no temperature-solids concentration interac-tion. This may not be true when temperature affects the solids structure and parti-cle size in the suspension, e.g. by hydrolysis of macromolecules, coagulation ofcolloids, and breakdown or buildup of agglomerates.

The energy of activation for viscous flow Ea is estimated at a constant shearrate, usually at 100 (1/s) (Saravacos, 1970; Rao, 1999). However, the structure ofsome food suspensions of high particle (pulp) concentration may be changed dueto this relatively high shear rate. For this reason, in such cases, a lower shear ratemay be preferable. Prentice and Huber (1983) used a shear rate of 10 (1/s) inevaluating the effect of temperature on the rheology of applesauce.


D. Dynamic ViscosityThe rheological characterization of complex semisolid foods requires, in ad-

dition to the shear viscosity 77, the dynamic viscosity 77", which is determined byoscillating instruments at various frequencies. Three basic (constitutive) equationsare used to simulate the experimental rheological data, i.e. Newton's, Hooke's,and Maxwell's (Kokini, 1992).

In dynamic rheological tests, the stress or strain is changed periodically (si-nusoidally) with time at a fixed frequency CD. The complex stresses and strains arerelated to the storage and loss moduli of the material, G 'and G "respectively.

The Bird-Carreau constitutive model, derived for polymer solutions, repre-sents the entire deformation of a material. Thus, the shear and dynamic viscosities(77 and 77') are estimated from the equations:

77'= Y —— —— (4-17)2

where 77^ is the shear viscosity of p element, y is the shear rate (1/s), co is the fre-quency of oscillation (1/s), and A/, /L2 are time constants (s).

When the shear and dynamic viscosities of food suspensions, like mayon-naise and margarine, are plotted against shear rate or oscillation frequency on log-log scales, parallel straight lines with negative slopes are obtained (Kokini, 1992).

In rheology, the dimensionless Deborah number (De) is an index of the fluidor solid behavior of the material. It is defined as the ratio of the time of deforma-tion tD over the time of observation tm and it can be estimated from the viscosity ofthe continuous phase 77,,, the shear rate y, and the plateau storage modulus G0:

De = (tD/t0) = (rlsy/G0) (4-18)

Liquid-like behavior is characterized by small De numbers, while large Denumbers correspond to solidlike materials (Rao, 1999).

74 Chapter 4

III. VISCOMETRIC MEASUREMENTS

A. Viscometers

The shear viscosity of Newtonian fluids and the rheological constants of thenon-Newtonian materials are determined using several instruments, which basi-cally measure the shear stress versus shear rate at a given temperature. The exter-nal stress (force/area) is applied at steady state, usually stepwise, either in ascend-ing or descending order. The stress takes the form of pressure drop in a capillarytube, or mechanical torque in a rotational system. The resulting rheogram is util-ized to extract rheological data for the material under the specific measurementconditions.

Three common types of viscometers are used in most fluid food applica-tions, i.e. the tube/capillary, the rotational, and the cone-and-plate viscometers(Figure 4.5). Some other types of instruments are used in special cases, e.g. theparallel plate, the slit, and the falling ball viscometers. Details on viscometers areprovided by the manufacturers of scientific instruments. Lists and descriptions ofviscometers can be found in the literature of rheology, e.g. van Wazer et al.(1963), Whorlow (1980), Bourne (1982, 1992), and Rao (1999).

In food process control, special sensors are used to monitor and control theviscosity of a material, if changes of this property are very important during proc-essing. A typical example is the hot wire viscometer, which can be used to monitormilk clotting in cheese tanks, or enzymatic hydrolysis of starch (Sato et al., 1990).Its operation is based on the change of heat transfer rate on the surface of a hotwire, caused by a viscosity change.

1. Capillary Tube ViscometerThe tube viscometer (Figure 4.5a) is based on measuring the flow rate (shear

rate) of a fluid in a capillary or small diameter smooth tube at constant pressuredrop (shear stress). For a Newtonian fluid, flowing through a straight tube of circu-lar cross section with diameter (D, m) and length (L, m) under a pressure drop(AP, Pa), the Poiseuille equation is applicable:

D(AP) /4L = ri(32Q/ic D3) (4- 1 9)

where Q is the volumetric flow rate (mVs) and rj is the viscosity (Pa s). By com-paring Eqs. (4-19) and (4-1) it follows that the shear stress rand shear rate yforNewtonian fluids are given by the relationships:

T = D (AP) /4Landy=32Q/ (n D3) (4-20)


AP

Q

a. Tube Viscometer

b. Rotating Coaxial Viscometer

<-TO

->

c. Cone-and-Plate Viscometer

Figure 4.5 Common viscometers a) capillary/tube; b) rotating coaxial; c) cone-and-plate.

76 Chapter 4

For non-Newtonian fluids, the shear stress is given by the same relationship,but the shear rate is a function of the rheological constants K (Pa sn) and n:

Y = [(l+ 3n) / 4n][ 32Q / (nD3)] (4-21)

Therefore, the power-law model for capillary tube flow becomes:

D(AP)/4L =K[(l+3n)/4n]"[32Q/(nD3)]" (4-22)

Equations (4-19) and (4-22) apply to the laminar flow in tubes, i.e. when theReynolds number (Re) is lower than 2,000. For flow of Newtonian fluids in circu-lar tubes of diameter (D, m), Re = (D u p)/r/, where u is the mean velocity (m/s),p is the density (kg/m3), and 77 is the viscosity (Pa s).

For power-law and Herschel-Bulkley fluids, the (Re) number is calculatedfrom the equation

Re = [(D"u -'" p)/(8n~1 K)] [ (4n) / (3n + 1)] (4-23)

The velocity profiles of non-Newtonian flows in a tube viscometer can bemeasured using nuclear magnetic resonance (McCarthy et al, 1992). The profileof CMC solutions was as predicted by the laminar flow equations, while the pro-file of tomato juice showed a significant deviation.

The capillary tube viscometers require no calibration, since the rheologicalconstants can be determined from fundamental flow equations. They are suitablefor both Newtonian and non-Newtonian fluids, and they can handle suspensions ofrelatively large particles. However, they require more time for preparation andmeasurement than the rotational viscometers, which are preferred for routine ap-plications. In tube viscometers, very high shear rates at the wall can be obtained,when small tube diameters are used. Slip flow near the wall may be a problemwith fluid suspensions of large particles.

2. Rotational ViscometersThe most common rotational viscometer is the Couette or concentric cylin-

der rheometer (Figure 4.5b).For Newtonian fluids the viscosity rj is given by the empirical equation

(Rao, 1999)

jj = CM/n (4-24)

where M is the applied torque (N m), /2is the rotational velocity (1/s) and C is theinstrument constant (1/m ), which, for the concentric cylinder system, becomes:


C = (l/4xh)[(l/ri)2-(l/rQ)2J (4-25)

The shear rate of non-Newtonian fluids in rotational viscometers is esti-mated from empirical equations as a function of the rotational velocity Q, the ratioof the radii (rt/r^, and the flow behavior index n.

In concentric cylinder systems, narrow gaps, e.g. (rt / rj = 0.95 are requiredto insure laminar flow. Fluid foods, containing large suspended particles, cannotbe measured in narrow-gap viscometers, and other instruments are more suitable,e.g. capillary tube.

Simpler rotational viscometers, e.g. rotating plates, bobs or spindles in rela-tively large fluid volumes, are used in many routine measurements of Newtonianfluids. Such instruments are not suitable for non-Newtonian fluids, since the ap-plied shear rate cannot be estimated accurately. However, in some systems, em-pirical relationships of rotational velocity Q and shear rate y can be used to con-struct the rheogram (T vrs. f) of the non-Newtonian fluid.

3. Cone-and-Plate ViscometerThe principle of cone-and-plate viscosimeter, or Weissenberger rheogo-

niometer, is shown in Figure 4.5c. This instrument is suitable for non-Newtonianfluid suspensions, containing small particles. The shear stress rand shear rate y arecalculated from the equations

T =3M/r0A (4-26)

(4-27)

where M (N m) is the torque, r0 (m) is the radius of the plate, and A (m2) is thearea of the cone.

For measurements on non-Newtonian fluids, the cone/plate angle should besmall (0 < 5°), usually 2-3°. Small samples should be used with this instrument.

In both rotational and cone-and-plate viscometers the shear rate is usuallyvaried stepwise, either in increasing (ascending) or decreasing (descending) order.

78 Chapter 4

B. Measurements on Fluid FoodsIn rheological measurements of fluid foods, significant variations are ob-

served on the viscometric data, which may be due to calibration of the instrumentor to changes in the structure of the food material.

Accurate calibration of instruments is essential for non- Newtonian andtime-dependent fluid foods (Sherman, 1984; Bourne, 1992). Standardization of theinstrument and the measuring procedure is necessary. Particular attention shouldbe paid to fluid dispersions, whose structure may change remarkably (breakdownor agglomeration) during sample preparation and transfer to the measuring system.

Dispersions of fluid foods develop a complex structure, depending on thesize, shape and concentration of the suspended particles. When subjected to in-creasing shear or stress, the suspensions may break down gradually, especially atlow shear stresses. These changes are caused by breakdown of particle agglomer-ates and untangling of non-spherical particles.

Many fluid foods do not obey the empirical non-Newtonian models over theentire range of shear rate, available in the commercial instruments. Therefore, it isimportant to specify the data and model range, which, preferably, should be simi-lar to the range of application in process engineering or product quality.

Temperature has a strong effect on the viscosity of Newtonian fluid foods,especially in highly viscous products, such as syrups, honey, and edible oils. Vis-cous heating during viscometric measurements of thick food fluids may result inerroneous low viscosities. In general, temperature has little effect on the flow be-havior index n of non-Newtonian fluid foods. The effect of temperature on theconsistency coefficient K is similar to the effect on the viscosity of Newtonianfluids, obeying the Arrhenius equation. However, the energy of activation for vis-cous flow Ea in non-Newtonian fluids is considerably lower than in Newtonianmaterials. In several food suspensions and pulps, Ea may be lower than 14.4kJ/mol, which is the energy of activation for viscous flow of pure water. This is anindication that the consistency coefficient K is affected more by the structure thanby the temperature of the suspension

The problems of measuring the rheological properties of liquid foods wereinvestigated in the European cooperative research project, COST 90 (Prentice andHuber, 1983). Eight laboratories from different countries participated in the study,using exclusively concentric-cylinder and cone-and-plate viscometers. Some ofthe results are analyzed and discussed, in relation to literature data, in Sections IVand V of this chapter.


IV. RHEOLOGICAL DATA OF FLUID FOODS

Due to the complexity of food composition and structure, rheological datashould be obtained by standardized measurements on each specific material. Thetechnical literature contains a wide variety of such data, and new results continueto be reported in recent years. Collections of rheological data of fluid foods havebeen published by Steffe et al. (1986), Kokini (1992), Urbicain and Lozano(1997), and Rao (1999). This critical review includes typical viscometric datafrom a wide variety of fluid foods, classified as follows: edible oils, sugar solu-tions and clarified juices, suspensions of plant materials, and emulsions and com-plex suspensions.

A. Edible OilsThe edible or cooking oils and fats are generally considered as Newtonian

fluids, if they do not contain particles or emulsions, which may enter the oil phaseduring processing and use. Collaborative measurements on a commercial cookingoil (mixture of soybean and rapeseed) showed a slight deviation from Newton'slaw (Prentice and Huber, 1983). Regression analysis of the rheological data, ob-tained in seven laboratories, using coaxial-cylinder and cone-and-plate viscome-ters at 25°C, resulted to the following equation:

log (r) = -1.2766 + 0.99673 log (y) (4-28)

where the rheological constants are K = 0.052 Pa s" and n = 0.99673.According to this equation, the apparent viscosity (r\a = K /"'') of the oil de-

pends slightly on the shear rate, for example r/a = 52 mPa s at 7= 1 (1/s), and r/a =51mPasat/=1000(l/s) .

Temperature has a strong effect on viscosity, especially of viscous oils. TheArrhenius equation yielded an energy of activation Ea = 28.3 kJ/mol in the tem-perature range 25-50°C for the oil described by Eq. (4-28). Empirical equations ofviscosity as a function of temperature for specific oils have been proposed in theliterature (Rao, 1999). Typical viscosities of edible oils (Kokini, 1992; Rao, 1999)are given in Table 4.1.

Table 4.1 Typical Viscosities of Edible OilsEdible oilCorn oilSoybean oilOlive oilCocoa butter

Temperature, °C23.923.940.025.0

Viscosity, mPa s52.354.336.3124.1

80 Chapter 4

B. Aqueous Newtonian Foods

]. Sugar Solutions and Clarified JuicesAqueous food fluids, including sugar solutions, honey, and clarified juices

(containing no paniculate matter) are considered, in general, Newtonian liquids.However, the COST 90 collaborative measurements on 50°Brix sucrose solution at25°C (Prentice and Huber, 1983) showed a slight non-Newtonian behavior with anestimated flow behavior index n = 0.986 and a consistency coefficient K = 7.4mPa s". This corresponds to an apparent viscosity rja = 6.9 mPa s at a shear rate of

Sugar solutions and clear juices are characterized by sharp decrease of vis-cosity at higher temperatures, according to the Arrhenius equation. Concentrationhas a strong positive effect on viscosity, and the combined tempera-ture/concentration effect can be expressed in most cases by empirical exponentialequations, similar to the generalized Eq. (4-15).

Figure 4.6 compares the viscosities of clarified apple juice (Saravacos,1970) and sucrose solutions (Perry et al., 1984) at three different temperatures, 20,40, and 60°C. It appears that the viscosities of the two fluids are similar, except atconcentrations higher than 60°Brix, where the apple juice becomes more viscous,probably because of the increased concentration of nonsugar solids.

Figure 4.7 shows that the viscosity of concord grape juice tends to be higherthan the viscosity of the corresponding sucrose solution, particularly at high con-centrations. This difference may be caused by the precipitation of some com-pounds of grape juice, which become insoluble at higher concentrations, for ex-ample tartrates. These compounds are known to precipitate during the storage ofgrape juices and wines at low temperatures.


1000^±ES:^3^di=Ei^:S3^t• apple juice

, . ,,,,• sucrose solutions

VI«CM

uwi>

o.i

Figure 4.6 Comparison of viscosities of depectinized apple juice and sucrose solutions at20-60°C. (From Saravacos, 1970, and Perry et al., 1984.)

1000

~.( • grape juice• sucrose solutions

Figure 4.7 Comparison of viscosities of Concord grape juice and sucrose solutions at40°C. (From Saravacos, 1970, and Perry et al., 1984.)

82 Chapter 4

Figure 4.8 shows the energies of activation for viscous flow of sucrose solu-tions (Perry et al., 1984), grape juice (Saravacos, 1970), and apple juice (Rao,1999) as a function of concentration.

The energies of activation Ea of the two juices are close to those of sucrosesolutions, increasing sharply at higher sugar concentrations. The extrapolated en-ergy of activation at 0°Brix corresponds to the Ea value of water (14.5 kJ/mol),which was estimated from viscosity-temperature data of the literature (Perry et al.,1984)

The energy of activation for viscous flow increases as the molecular weightof the dissolved sugar is increased. The viscosity ij of a sugar solution can be ex-pressed by an empirical model, analogous to the Arrhenius equation (Chirife andBuera, 1994):

77 = a exp (EX) (4-29)

where X is the mole fraction of the sugar, a is a constant close to 1, and £ is acharacteristic constant, similar to the Arrhenius activation energy. Calculated val-ues of constant E are: 27.93 for glucose, 57.19 for sucrose, and 94.46 for cornsyrup of 35.4 DE (dextrose equivalent).

T'\ sucrose solutionsgrape juice

™ f-^ • apple juice

Figure 4.8 Activation energies for viscous flow of sucrose solutions (Perry et al., 1984),grape juice (Saravacos, 1970), and depectinized apple juice (Rao, 1999).


Molasses (thick liquid residues of sugar manufacturing) have high viscosi-ties, e.g. 6600 to 374 mPa s in the temperature range 21-66°C (Hayes, 1987), fromwhich an activation energy of 51.3 kJ/mol was estimated.

Clarified cherry juice behaves as a Newtonian fluid in the range 22-74°Brixand 5-70°C with energies of activation 14.4-61 kJ/mol (Giner et al, 1996). Thecombined effect of temperature and concentration on viscosity is described wellwith an empirical exponential model similar to Eq. (4-15). Similar activation ener-gies (26.6-64.4 kJ/mol) were obtained in clarified crab (small) apple juice (Cepedaet al., 1999). Clarified orange juice behaves as a Newtonian fluid in the range 30-63.5°Brix with energies of activation Ea in the range 17.7-40 kJ/mol (Ibarz et al.,1994). The relatively low Ea values may be due to the presence of colloidal parti-cles in the clarified juice.

Similar rheological behavior is shown by other clarified fruit juices: appleand pear juices and concentrates (Ibarz et al., 1987), peach juices and concentrates(Ibarz et al., 1992a), and black currant juices (Ibarz et al., 1992b). Clarified bananajuice (20-79.7°Brix) has viscosities ranging from 1 to 6000 mPa s in the range 30-70°C, and energies of activation for viscous flow 25-78 kJ/mol (Khalil et al.,1989).

2. Honey and Sugar ExtractsThe viscosity of honeys depends strongly on the concentration of soluble

solids (°Brix), temperature, and plant origin. Most honeys have a concentrationbetween 75-83°Brix, and their viscosity at 20°C is in the range of 4-20 Pa s. Tem-perature has a strong effect, with corresponding high energies of activation Ea.Utilizing the viscosity-temperature data (10-80°C) of five types of honey, pub-lished by Rao (1999), Ea values of 65-70 kJ/mol were obtained, which are inagreement with the generalized diagram of Figure 4.8. Chinese honeys (Junzhengand Changying, 1999) have lower sugar concentrations (71.5-80.2°Brix) and vis-cosities (0.33-6.30 Pa s at 20°C) with estimated activation energies 60-65 kJ/mol.Australian honeys have high sugar concentrations (82.4-83.3°Brix) with high vis-cosities (about 100 Pa s at 10°C), and a strong effect of temperature (Bhandari etal., 1999).

Licorice extract of soluble solids concentration 3-50°Brix exhibited Newto-nian behavior in the temperature range of 10-60°C, with energies of activation14.4-61 kJ/mol (Maskan, 1999).

The viscosity data of honey, presented by Rao (1999), and molasses (Hayes,1987) are plotted in Figure 4.9, snowing a nearly exponential drop of viscositywith increasing temperature.

84 Chapter 4

0.1 -L

30 40 50Temperature (°C)

Figure 4.9 Viscosities of honeys and molasses.

3. Other Clear SolutionsThe viscosity of ethanol at 20°C is 1.2 mPa s, and that of glycerol is 1490

mPa s (Lewis, 1987). The viscosity of wines is higher than that of water, rangingfor 1.3-1.67 mPa s for ordinary wines at 25°C, and 1.95-2.45 mPa s for fortifiedwines, containing glycerol (Rao, 1999). The energy of activation for viscous flowof wines is in the range of 18.8-28.2 kJ/mol, i.e. significantly higher than that ofpure water.

The viscosity of fermenting musts is higher than that of wines, due to thepresence of dissolved sugars and suspended particles in the liquid (Lopez et al.,1989), For white musts, the viscosity varies in the range of 3-4 mPa s at 14-22°C,with activation energy about that of water (14.4 kJ/mol). The viscosity of beer isclose to that of wine, i.e. 1.3 mPa s at 20°C (Lewis, 1987). The viscosity of salt(sodium chloride) solutions and brines, depends on the concentration of the ionicspecies, for example 2.7 mPa s at 22% salt and 2°C (Hayes, 1987).


C. Plant Biopolymer Solutions and SuspensionsPlant hydrocolloids (gums and starches) are components of many fluid

foods and they are responsible for their non-Newtonian behavior. They are pre-sent, in relatively low concentrations, as molecular solutions, colloidal dispersions,and particulate suspensions. Dispersions of plant gums contain macromolecules ofcoil configurations, while starch suspensions contain granular particles.

Collaborative measurements, within the COST 90 European project, weremade on solutions of guar gum, carrageenan gum, xanthan gum, and Karaya gum,using coaxial-cylinder and cone-and-plate viscometers. The results showed highconsistency coefficients (K), some yield stresses (TO), and flow behavior indices(n) lower than (1), an indication of shear-thinning (pseudoplastic) behavior.

Published shear stress r versus shear rate ^data of 1% gum solutions (Pren-tice and Huber, 1983) were analyzed statistically, and the resulting regression linesare shown in the logarithmic plots of Figures 4.10 to 4.13. The Theological con-stants K and n of the power-law model (Eq. 4-8) are shown in Table 4.2:

The rheological properties of the gums depend strongly on the concentrationand type of macromolecules. Xanthan gums show high consistency and low flowbehavior indices, presumably due to the high molecular weight and the large col-loidal particles in the solution.

Table 4.2 Rheological Constants of 1% Food Gum Solutions at 25°CFood gum Consistency coefficient Flow behavior index

_________________K, Pa s"____________n______Guar 1.24 0.625Carrageenan 1.40 0.485Xanthan 9.21 0.183Karaya_____________1.23____________0.465

86 Chapter 4

1000

0.001

0.1

0.01

0.001 10 1000 100000

Shear Rate (1/s)

Figure 4.10 Rheological data of 1% guar gum solution at 25°C. (Data from Prentice andHuber,1983.)

1000Carrageenan gum

solution

0.10.001 0.1 10 1000

Shear Rate (1/s)

100000

Figure 4.11 Rheological data of 1% carrageenan gum solution at 25°C. (Data from Pren-tice and Huber, 1983.)


1000

0.001 0.1 10 1000

Shear Rate (1/s)

100000

Figure 4.12 Rheological data of 1% xanthan gum solution at 25°C. (Data from Prenticeand Huber, 1983.)

1000

0.1

0.010.001 10 1000 100000

Shear Rate (1/s)

Figure 4.13 Rheological data of 1% karaya gum solution at 25°C. (Data from Prenticeand Huber, 1983.)

88 Chapter 4

Tube viscometry of aqueous pectin solutions showed pseudoplastic behavior(Saravacos et al., 1967). Plots of shear rate rversus sear rate /for 2.5% and 5.0%pectin solutions at 25°C (Figure 4.14) show the following rheological properties:

K= 3,590 Pa sn and n = 0.767 for the 2.5% solution; K = 57,420 Pa sn

and « = 0.822 for the 5.0% solution.The power-law model applies to carboxy-methyl-cellulose (CMC) solutions

(0.5-1.5%) in the temperature range 25-125°C, which are of interest to asepticprocessing of foods. The rheological properties (K, n) were obtained using a sealedpressure rheometer (Vais et al., 1999). The consistency coefficient of the 1.5%solution decreased from 17 to 3.4 Pa sn, and the flow behavior index n remainednearly constant at 0.43 at temperatures 25-125°C.

Starch suspensions and solutions exhibit complex rheological properties,due to the diverse particulate and molecular composition and the physicochemicalchanges, induced by temperature (gelatinization) and molecular interactions withother food components, like proteins. Rheological characterization of starch sys-tems requires not only shear stress-shear rate data, but also viscoelastic measure-ments and particle and biopolymer characterization (Rao, 1999).

1000 -,

aa-

- 100

10

4 Pectin— 2.5%— 5.0%

10 100

Shear Rate, 1/s

1000

Figure 4.14 Rheological data of aqueous pectin solutions (From Saravacos et al., 1967.)


D. Cloudy Juices and PulpsThe presence of hydrocolloids (gums) and suspended particles (e.g. gran-

ules or fibers) in fluid plant foods changes the rheological properties into non-Newtonian, usually pseudoplastic fluids.

Aqueous suspensions of dietary fibers, from orange or peach, exhibit strongnon-Newtonian behavior, particularly at concentrations higher than 5%(Grigelmo-Miguel et al., 1997). The energy of activation for viscous flow is low,3-18.4 kJ/mol, and the effect of concentration on the consistency coefficient of thesuspension is higher than that of temperature.

Rheological measurements on applesauce, using a capillary tube viscometer,yielded the values K = 12.7 Pa sn, and n = 0.28 (Saravacos, 1968). The consistencycoefficient K decreased from 12.7 to 0.49 Pa sn and the flow behavior index n in-creased slightly from 0.28 to 0.35, when the pulp content was reduced by passingthe applesauce through an 0.04 inch (1 mm) screen. Similar results on therheological constants of applesauce were obtained in the COST 90 cooperativemeasurements (Section V of this chapter).

Addition of sugar (glucose) to banana pulp reduces the apparent viscosityand increases the temperature dependence of the juice. The activation energy in-creased from 8 to 54 kJ/mol when the sugar concentration increased from 21 to51°Brix (Guerrero and Alzamora, 1997). A similar change of the rheological prop-erties and activation energy by glucose addition was observed on peach, papayaand mango purees (Guerrero and Alzamora, 1998). The combined effect of con-centration and temperature follows the generalized exponential model of Eq. (4-15).

Cloudy apple juice becomes a non-Newtonian fluid at concentrations higherthan 40°Brix, with flow behavior indices n = 0.65-0.80, and activation energieslower than the clarified juice, e.g 25.5 kJ/mol versus 35 kJ/mol at 50°Brix (Sara-vacos, 1970). A similar rheological behavior was observed on raspberry juice,which changed from Newtonian to non-Newtonian above 25°Brix, with « = 0.90-0.60 (Ibarz and Pagan, 1987).

Ultrafiltration of orange juice removes part of the particles and hydrocol-loids (pectin), decreasing the consistency coefficient K, and increasing the flowbehavior index n and the activation energy (Hernandez et al., 1995). This treat-ment increases significantly the heat transfer coefficient in the evaporation of or-ange juice. Shear-thinning food fluids have improved heat transfer characteristicsin agitated kettles and heat exchangers, since their apparent viscosity is reducedconsiderably by agitation (Saravacos and Moyer, 1967).

The power-law model of non-Newtonian fluids has been applied to varioussuspensions of starch and pulp materials: cooked debranned maize flour with flowbehavior indices n = 0.258-0.668 (Bhattacharaya and Bhattacharaya, 1996); rice-blackgram suspensions (28-44%TS) increased the (n) value from (0.3-0.4) to(0.45-0.50) by fermentation, evidently due to a breakdown of biopolymers (Bhat-tacharaya and Bhat, 1997).

90 Chapter 4

Enzymatic breakdown of starches and other biopolymers results in lowerviscosities and higher n values, such as the enzymatic extrusion of rice starch(Tomas et al., 1997) and the enzyme treatment of mango pulp (Bhattacharaya andRestogi, 1998). The rheology of tomato juice and tomato concentrates has re-ceived special attention due to its importance in processing and in product quality.Typical theological properties are: K = 0.22 to 12.9 Pa sn and n = 0.59 to 0.41 for5.8 to 25.0% TS (coaxial- cylinder, Harper and El-Sahrigi, 1965); K = 0.22 to 52Pa s" and n = 0.581 to 0.177 for 5.6 to 32.6°Brix (tube viscometer, Saravacos et al.,1967).

The processing method has a significant effect on the rheology of tomatoconcentrates. The "hot break" product is more viscous than the "cold break" mate-rial. Thus, in the range 10-25°Brix, the lvalue of the "cold break" tomato concen-trates decreased from (5-80 Pa sn) to (4-30 sn), and the activation energy from 22.7to 17.0 kJ/mol, compared to the "hot break" product (Fito et al., 1983).

The particle size and pulp concentration strongly affect the consistency coef-ficient K of tomato concentrates (Rao, 1999). An empirical rule is to scale the vis-cosity by a factor of (total solids)2'5.

E. Emulsions and Complex Suspensions

Oil/water (o/w) food emulsions, like mayonnaise and salad dressings, arenon-Newtonian fluids with high consistency coefficients K and flow behavior in-dices n lower than 1. Mayonnaise contains 70-80% oil particles (droplets) in thesize range 0.01-10 um, dispersed in an aqueous phase. Food emulsions are stabi-lized by adsorption of biopolymers (gums, proteins, lecithin) on the particle sur-faces (Rao, 1999). Because of their structure, emulsions may exhibit time-dependent theological properties, as well as viscoelastic behavior.

Addition of sugar to emulsions changes significantly their theological prop-erties. Thus, by adding 8% sugar to sunflower oil-water emulsions, the consistencycoefficient K of the power law model decreased from (2.6-3.6) to (0.6-2.2), whilethe flow behavior index n remained nearly constant (n = 0.49). The activation en-ergy Ea decreased from 31 to 10.7 kJ/mol (Maskan and Gogus, 2000).

Milk may be considered as dilute emulsion of fat globules in an aqueoussolution of lactose and other components. Single strength (non concentrated) milkbehaves as a Newtonian fluid with a viscosity higher than that of water (Kokini,1992). Literature data on homogenized milk show viscosities decreasing from 3.4to 0.6 mPa s, in the temperature range 0-80°C (Figure 4.15).

Concentration of milk by evaporation above 22.3%TS changes the rheologyto non-Newtonian (n = 0.89 at 42.4%TS), with a sharp increase of the activationenergy from 10 to 49 kJ/mol (Velez-Ruiz and Barbosa, 1998). The combined ef-fect of temperature and concentration on the consistency coefficient K follows thegeneralized model of Eq. (4-15). Freeze-concentrated milk showed a similar


rheological behavior (Chang and Hartel, 1997). The power-law model was foundapplicable, with the flow behavior index decreasing from 1.0 to 0.89, and the acti-vation energy increasing from 20 to 60 kJ/mol in the range of 20-40%TS.

Sodium caseinate suspensions behave as Bingham plastic materials, follow-ing an empirical model similar to Eq. (4-15). The activation energy of casein sus-pensions increased from 14.6 to 37.7 kJ/mol in casein concentrations 10-16%TS.Buttermilk shows a thixotropic (time-dependent) behavior. The Herschel-Bulkleymodel can be applied, using a structural correction factor (Butler and McNulty,1995).

The flow properties of yogurt can be expressed by the power-law or theHerschel-Bulkley model. However, since yogurt is a complex material, exhibitingboth fluid and semisolid (gel) properties, time-dependency (thixotropy) and vis-coelasticity should be considered (Afonson and Maia, 1999). A strong degradationof the yogurt may take place at high shear rate. The activation energy for viscousflow at temperatures higher than 25°C increases sharply from 24 to 65 kJ/mol,evidently due to action of thermophilic clotting bacteria (Benezech and Maingon-nat, 1994).

The rheological properties of salmon surimi can be expressed by either theHerschel-Bulkley or the Casson models, with the flow behavior index n in therange of 0.58-0.75 (Bourami et al., 1997).

Table 4.3 shows typical rheological properties of food emulsions.

10 T-

20 40 60

Temperature (°C)

80 100

Figure 4.15 Viscosity of homogenized milk. (Data from Kokini, 1992.)

92 Chapter 4

Table 4.3 Typical Rheological Data of Food Emulsions at 25°C________Food emulsion Consistency coefficient (K) Flow behavior index (n)

Pas"MayonnaiseMustardMargarinePeanut butterCream, 40% fat

6.418.5

297.6501.1

0.0069 (40°C)

0.550.39

0.0740.065

1.0Source: From Kokini, 1992.

V. REGRESSION OF RHEOLOGICAL DATA OF FOODS

Most of the literature data on the viscosity or Theological constants of foodsare available in the form of tables or diagrams at specific temperatures andconcentrations. More information can be gained by analyzing the data statistically,and extracting a regression line, typical of a food product, or a group of similarproducts. However, statistical analysis of the literature data is difficult, because themeasurements were made with different instruments on different products, and thestatistical samples are often small.

Rheological data obtained in the COST 90 cooperative measurements (Pren-tice and Huber, 1983) are suitable for statistical analysis, and some regressionlines were presented in Section (IV.C) of this chapter. Additional regression linesare presented in this section for applesauce and chocolate.

Approximate representative lines can be obtained by non-linear regressionof several data on a food material, obtained by different investigators, providedthat the statistical sample is large enough. Details of this regression technique aregiven in Chapter 6.

From a review of the literature, rheological data on the following food prod-ucts were found suitable for regression analysis: edible oils, applesauce, tomato,orange, pear, and mango juices and concentrates, and chocolate.

A. Edible Oils

Literature data on various edible oils and fats were obtained from Kokini(1992), Noureddini et al. (1992), and Rao (1999). The oils are assumed to be New-tonian fluids. The reported viscosities ranged from 1.6 to 451 mPa s at tempera-tures from 0 to 121°C. The viscosities are plotted versus the temperature in Figure4.16.

The following form of the Arrhenius equation was applied for the calcula-tion of the activation energy for viscous flow Ea:


77 = 77 exp[—-(— -——)]/ / . ^ ^ » (4-30)

where 77 and TJO are the viscosities at temperatures T and T0, respectively, and R isthe gas constant (8.314 J/mol K).

For a reference temperature T0 = 25°C, the estimated viscosity of an "aver-age" oil is 7j0 = 55 mPa s, and the activation energy Ea = 45 kJ/mol. These valuesare comparable to the values (77 = 52 mPa s and Ea = 53.2 kJ/mol) obtained on atypical edible oil, in the cooperative COST 90 project (Prentice and Huber, 1983).

1000 TZ

a.s

20 40 60 80 100

Temperature (°C)

120 140

Figure 4.16 Regression line of viscosity data of edible oils.

94 Chapter 4

B. Fruit and Vegetable Products

1. ApplesauceRheological data (shear stress versus shear rate) were obtained in seven

laboratories on applesauce of 22.5°Brix (screen opening 0.6 mm) at 25°C (Prenticeand Huber, 1983). The regression line of the power-law model yielded the follow-ing rheological constants: K= 26 A Pa s" and n = 0.286 (Figure 4.17).

1000I Apple Sauce |

10.001 0.01 0.1 1 10 100 1000

Shear Rate (1/s)

Figure 4.17 Rheological data of applesauce at 25°C. (Data from Prentice and Huber,1983.)


2. Fruit /Vegetable Juices and ConcentratesRheological data on tomato, orange, pear, and mango juices and concen-

trates, suitable for regression analysis, were obtained from Harper and El-Sahrigi(1965), Fito (1983), Manohar et al. (1991), and Rao (1999).

The fruit and vegetable juices/concentrates, considered in this section, areassumed to behave as non-Newtonian fluids, following the power-law orHerschel-Bulkley models. The consistency coefficient K was estimated from thefollowing form of the generalized Eq. (4-15), Krokida et al. (2000):

(4-31)

The flow behavior index n is assumed to be a linear function of concentra-tion and independent of temperature, according to the equation:

n = n0-bC (4-32)

where K = K0 when C = 0 and T = To, n = n0 when C = 0. C is the concentrationand B, b are constants.

The regression lines for the four products are shown in Figures 4.18 to 4.21.The consistency coefficient K increases exponentially with the concentration,while the flow behavior index n decreases slightly. Concentration has a strongereffect on K of tomato than the other three fruits. The higher activation energy fororange and mango may be due to the higher sugar content.

Table 4.4 Estimated Values of the Parameters of Eqs. (4-31) and (4-32)Material

TomatoOrange

PearMango

K0. Pa s"1.279.282.151.85

B0.1490.0770.0870.089

Em kJ/mol15.835.016.132.1

n00.4030.9500.3480.332

b0.00280.00340.00000.0017

(Q C 3 00 o_ o OQ o'

o

Flow

Beh

avio

r In

dex

(-)C

onsi

sten

cy C

oeffi

cien

t (Pa

s")

o o

n o

H o to

O J u T3 ff

(Q C 3 to

Flow

Beh

avio

r In

dex

(-)

C 8

p o

Con

sist

ency

Coe

ffici

ent (

Pa s"

)o o

\.

n o 3 O o> 3 a o 3 8-

\ \

y\ \-a I c

o o_ o (Q o' D> TJ CO a fl>' in O O o a. u>

98 Chapter 4

cM« 100 -

e

J> 10 -cu>e£Ift

ou

n 1 .

_^*

*^^-^^^ -^^^^^ix^^^"*^ I

-^^ •

^^ ':.^"^^^^'^

^<^c l

———

.-*'^,

^^^*^» ————r :=—

Temperature (°C)

—————— • 40 ———————— • 60

20 40 60

Concentration (% solids)

•8_oRUoa

i.o

20 40

Concentration (% solids)

60

Figure 4.20 Rheological data of pear juice and concentrates.

(Q C 3 k) po n

Flow

Beh

avio

r In

dex

(-)C

onsi

sten

cy C

oeffi

cien

t (P

as )

P o

Q n sa I 8

65 1

O o a a o a

<D O_ O (Q o' 0) O •o <D to' to C OL -n o o Q.

(0 to to

100 Chapter 4

C. Chocolate

The Theological data on chocolate at 40°C, obtained in the COST 90 project,were analyzed, assuming that the Casson model (Eq. 4-10) is applicable. The re-gression line, shown in Figure 4.22, gave the following Theological constants: TO =

0.527.0 Pa and Kc = 1.68 (Pa s )™ .

10000 -p=

1000IM

CO

'I

0.001 0.01 0.1 1 10

Shear Rate (1/s)

100 1000

Figure 4.22 Rheological data of chocolate at 40°C. (Data from Prentice and Huber,1983.)


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Benezech, T., Maingonnat, J.F. 1994. Characterization of the Rheological Proper-ties of Yogurt. A Review. J Food Eng 21:447-412.

Bhandari, B., D'Arcy, B., Chow, S. 1999. Rheology of Selected Australian Hon-eys. J Food Eng 41:65-68.

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Butler, F., McNulty, P. 1995. Time-Dependent Rheological Characterization ofButtermilk at 5°C. J Food Eng 25:569-580.

Cepeda, E., Villaran, M.C. 1999. Density and Viscosity of Malus Floribunda Juiceas a Function of Concentration and Temperature. J Food Eng 41:103-107.

Chang, Y.-H., Hartel, R.W. 1997. Flow Properties of Freeze-Concentrated SkimMilk. J Food Eng 31:375-386,

Chirife, J., Buera, M.P. 1994. A Simple Model for Predicting the Viscosity ofSugar and Oligosaccharide Solutions. J Food Eng 33:221-226.

Fichtali, J. 1993. A Rheological Model for Sodium Caseinate. J Food Eng 19:203-211.

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Giner, J., Ibarz, A., Garza, S., Xhian-Quan, S. 1996. Rheology of Clarified CherryJuice. J Food Eng 30:147-154.

Grigelmo-Miguel, N., Ibarz-Ribas, A., Martin-Belloso, O. 1997. Flow Propertiesof Orange and Peach Dietary Fiber. IFT 97. Orlando, FL, paper No. 35A-2.

102 Chapter 4

Guerrero, S.M., Alzamora, S.M. 1997. Effect of pH, Temperature and GlucoseAddition on Flow Behavior of Fruit Purees. I. Banana Puree. J Food Eng33:239-256.

Guerrero, S.M., Alzamora, S.M. 1998. Effect of Temperature and Glucose Addi-tion on Flow Behavior of Fruit Purees: Peach, Papaya and Mango Purees. JFood Eng 37:77-101.

Harper, J.C, El-Sahrigi, A.F. 1965. Viscometric Behavior of Tomato Concentrates.J Food Sci 30:470-476.

Hayes, G.D. 1987. Food Engineering Data Handbook. London: Longman Scien-tific & Technical.

Hernandez, E., Chen, C.S., Johnson, J., Carter, R.D. 1995. Viscosity Changes inOrange Juices after Ultrafiltration and Evaporation. J Food Eng 25:387-396.

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Ibarz, A., Pagan, J. 1987. Rheology of Raspberry Juices. J Food Eng 6:269-289.Ibarz, A., Gonzalez, C., Esplugas, S. 1994. Rheology of Clarified Fruit Juices. III.

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Rheological Behavior of Clarified Banana Juice: Effect of Temperature andConcentration. J Food Eng 10:231-240.

Kitano, T., Kataoka, T., Shirota, T. 1981. An Empirical Equation of the RelativeViscosity of Polymer Melts Filled with Various Inorganic Fillers. RheologicalActa 10:207-209.

Kokini, J.L. 1987. The Physical Basis of Liquid Food Texture and Texture-TasteInteractions. J Food Eng 6:51-81.

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Krokida, M.K., Maroulis, Z.B., Saravacos, G.D. 2000. Rheological Properties ofFluid Fruit and Vegetable Products: Compilation of Literature Data. Int JFood Properties, in press.

Lewis, M.J. 1987. Physical Properties of Foods and Food Processing Systems.London: Ellis Horwood,

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Manohar, B., Ramakrishna, P., Udayashankar, K. 1991. Some Physical Propertiesof Mango Pulp Concentrate. J Texture Studies 21:179-190.

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Transport of Water in Food Materials

I. INTRODUCTION

The transport of water in food materials is of fundamental importance toseveral food processing operations, as well as to various physical, chemical andmicrobiological changes of food products.

Water transport within the food material is the main rate-controlling mecha-nism in drying operations (Keey, 1972), and various techniques have been devel-oped to increase the drying rate, resulting in reduction of the cost of drying, andimprovement of food quality. Water transport to and from food products duringstorage is important in controlling food preservation and food quality.

Although various mechanisms have been proposed to explain water trans-port in food materials, the diffusion model yields satisfactory results for engineer-ing and technological applications. The water chemical potential model appears tobe more appropriate for cellular foods (Gekas, 1992), but the required physical andchemical properties are difficult to determine and they may change during meas-urement.

The mechanisms of water transport in solids are reviewed in Section II, withemphasis on the diffusion in polymers, which constitute the structural backbone offood materials, and for which basic literature is available (Vieth, 1991). The meth-ods of measurement and calculation of mass diffusivity are discussed in detail inSection III.

Food structure plays a decisive role in water transport processes within thefood materials (Saravacos, 1998). Model food materials, based on granular andgelatinized starch, are convenient experimental materials in studying the mecha-nism of water transport in various food structures (Section IV). Characteristicmoisture diffusivities in the main classes of foods are given in tabular form in Sec-tion V. A more detailed and unified regression analysis of the literature data onmoisture diffusivity in foods is presented in Chapter 6.

105

106 Chapters

II. DIFFUSION OF WATER IN SOLIDS

The diffusion of water in simple gases and liquids can be analyzed and pre-dicted by molecular dynamics and empirical correlations, as discussed in Chapter2. The transport of water and other small molecules in solids and semisolids is ofparticular importance to foods and food processing systems. The mechanism ofwater transport in solid materials is less well understood than in fluid systems, andempirical approaches are often used to estimate the transport properties.

The transport of water is of fundamental importance to the drying of solids(Keey, 1972). In the drying process, liquid water is removed first by a hydrody-namic gradient and capillary forces. As drying progresses, water is removed byvapor diffusion, and finally by desorption from the solids.

The transport of water in solids is usually assumed to be controlled by mo-lecular diffusion, i.e. the driving force is a concentration gradient (dC/dz) or theequivalent moisture content gradient (dX/dz). For simplified analysis and calcula-tions, one-dimensional diffusion is considered, and the Pick diffusion equation isapplied:

(5-1)dt dz 8z

The diffusion coefficient D of water in solids is usually defined as the effec-tive moisture diffusivity, which is an overall transport property, incorporating alltransport mechanisms.

In addition to diffusion, water may be transported by other mechanisms,such as hydrodynamic and capillary flow, depending on the structure of the solidmaterial.

The effective diffusivity D of a molecular species A in a porous solid ismuch lower than the diffusivity of A in a gas medium B, DAB, according to theequation (Geankoplis, 1993):

D = (s/T)DAB (5-2)

where sis the bulk porosity and ris the tortuosity of the solids (T> 1). The tortu-osity is a measure of the tortuous (complex) path of the diffusing molecules,T=Lg/L, where Le is the equivalent length of the diffusion path, and L is thestraight-line thickness of the sample.

Equation (5-2) is applied to catalysts and other solids of fixed structure, anda reliable tortuosity can be determined experimentally. However, determination oftortuosity in food materials is difficult because food structure changes substan-tially during food processing and storage. For this reason, the effective moisturediffusivity is determined directly by experimental techniques.

Transport of Water in Food Materials 107

Molecular diffusion is the prevalent mass transport mechanism in most foodmaterials. In molecular diffusion, the mean free path of the diffusing molecules ismuch shorter than the pore or capillary diameter, and most collisions are betweenthe molecules than with the walls.

In Knudsen diffusion, the mean free path of the molecules is near the size ofthe pore or capillary diameter, and the molecules collide more with the walls thanwith each other (Brodkey and Hershey, 1988).

The Knudsen diffusion coefficient DK is given by the equation

DK = 48.5d0(T/MA)"'2 (5-3)

where d0 is the pore (capillary) diameter, T is the temperature, and MA is the mo-lecular weight of the diffusing species.

Equation (5-3) indicates that the Knudsen diffusion coefficient DK in gasesis a function of the square root of temperature, in contrast to the molecular diffu-sivity D, which is proportional to the (3/2) power of the temperature equation (2-28).

Other diffusion mechanisms, which are of minor importance to food sys-tems at normal conditions, are:

• Surface diffusion (mass transport by surface concentration gradients)• Molecular effusion (passage of molecules through a small aperture in a thin

plate into a vacuum)• Thermal diffusion (mass transport due to a temperature gradient)

The capillaries in Knudsen diffusion are much longer than in molecular ef-fusion. Knudsen diffusion may be prevalent in capillary systems under vacuum, asin freeze-drying, where the mean free path of the molecules is very long.

A. Diffusion of Water in PolymersAn indication of the mechanism of water transport in polymers is the sorp-

tion kinetics test on a sample of the material. Gravimetric sorption data (SectionIII) in a polymeric sample (usually a film) are compared to the generalized sorp-tion equation (Peppas and Brannon-Peppas, 1994):

( M / M e ) = k f (5-4)

where M and Me are the moisture contents after sorption time t and at equilibrium,respectively, k is a constant, and n is the diffusion index.

The diffusion index characterizes the type of diffusion in the material:

• n = 0.5 Fickian diffusion• 0.5 < n < 1 non-Fickian diffusion• n = 1 type II diffusion

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In the case of Fickian diffusion, the constant k of Eq. (5-4) is related to thediffusivity D and the sample thickness L by Eq. (5-5), which is derived from Eq.(5-9):

k = 4(D/nl2) (5-5)

Thus, sorption kinetics data at a constant temperature can be used to deter-mine D.

The effective moisture diffusivity D depends strongly on the physical struc-ture of the polymeric material. Thus, values of D of the order of 10'14 are charac-teristic of the glassy state of foods, while D increases by nearly 1000 times abovethe glass transition temperature (rubbery state).

As water enters the polymeric network, mechanical stresses are developed,resulting in chain rearrangements and significant changes of the polymer structure.These changes are evidenced by characteristic swelling and relaxation phenomena,which affect substantially the transport mechanism and transport properties ofwater in the polymer. Swelling reduces the density of the polymer material and theglass transition temperature Tg, resulting in increased moisture diffusivity.

The relative importance of relaxation to diffusion is expressed by the diffu-sional Deborah number De:

De = A/t = ZD/L2 (5-6)

where /I is the characteristic relaxation time, t = L2/D is the characteristic diffusiontime and L is the film (slab) thickness.

The following diffusion types can be indicated by the Deborah number:

• De » 1, Fickian diffusion: relaxation time is much higher than diffusion time• De « 1, Fickian diffusion: very fast relaxation• De = 1, Case II diffusion : diffusion is controlled by molecular relaxation

Equation (5-6) indicates that the Deborah number is inversely proportionalto the square of sample thickness, i.e. the type diffusion mechanism dependsstrongly on the dimensions of the material. This dependence is evidenced in thedetermination of moisture diffusivity D by various techniques, when different val-ues of D are obtained for the same material. Thus, smaller D values are obtainedby the sorption kinetics technique, which uses thin firms, than by the drying ratemethod, which uses thicker samples. In the drying method, a higher porosity of thethick sample is obtained, increasing further the moisture diffusivity.


III. DETERMINATION OF MASS DIFFUSIVITY IN SOLIDS

The determination of mass diffusivity in solid and semisolid materials isessential for the quantitative analysis and control of several mass transfer opera-tions and applications, such as drying, adsorption, extraction, membrane separa-tions, ion exchange, and packaging. The methods of measurement and estimation,discussed in this Section are applicable to the diffusion of all small molecules(gases, vapors, and liquids) in solid or semisolid substrates. Of particular interestto food systems is the measurement and estimation of difrusivity of water, ormoisture diffusivity, since water, as a liquid or vapor, is involved in most foodprocessing operations and packaging/storage applications. There is no standardmethod for evaluating the effective diffusivity of water in food materials, due to thecomplex physical structure, and the changes that occur to the food samples duringthe measurement procedure.

It should be emphasized that most methods estimate the effective diffusivityD of water in the food material for a well-defined set of conditions. As defined ear-lier in this chapter, D is an overall transport property, incorporating all transportmechanisms for the particular process, i.e. liquid and vapor diffusion, Knudsendiffusion, capillary flow, hydrodynamic flow, etc. It is assumed that the transport ofwater is described by the diffusion (Pick) equation (5-1), and the driving force is anoverall concentration gradient, a simplification for the actual transport mecha-nisms, but nevertheless convenient for obtaining quantitative data for such com-plex systems. The use of chemical potential of water as a driving force, (Eq. 5-35),although thermodynamically sound, is difficult to apply in practice, and there islittle literature for food materials (Gekas, 1992).

Table 1 lists the most important methods of measurement and estimation ofthe effective moisture diffusivity, which are subsequently discussed and comparedin connection with their applicability to food materials (Zogzas et al., 1994b; Zog-zas and Maroulis, 1996; Saravacos, 1995). As a general rule, the method of meas-urement should be related to the actual intended application, for example the dryingmethod is recommended for applications related to drying, sorption kinetics is ap-plicable to moisture adsorption in food storage, and moisture distribution is relatedto mass transfer between contacted materials.

Some elegant techniques of determination of moisture diffusivity, such as thenuclear magnetic imaging (NMR) and the pulsed field gradient methods, have beenproposed in the literature (McCarthy et al., 1991, 1994), but have found little prac-tical application to food materials.

Mass diffusivity in fluid food systems (Cussler, 1997) is discussed in Chapter7.

110 Chapters

Table 5.1 Methods of Determination of Mass Diffusivity

1. Sorption Kineticsa. Gravimetric Methodb. Chromatographic Method

2. Permeabilitya. Time-Lag Methodb. Unsteady-State Method

3. Distribution of Diffusant

4. Drying Kineticsa. Constant Diffusivityb. Variable Diffusivity

1. Simplified Methods2. Simulation Method3. Numerical Methods4. Regular Regime Method

A. Sorption Kinetics

The sorption kinetics and permeability methods were developed and areapplied extensively in polymer science for the determination of mass diffusivity ofgases and vapors in solid materials. Samples in the form of thin slabs (films) arenormally used for transient adsorption or desorption at constant gas or vapor pres-sure and temperature.

1. Gravimetric MethodThe principle of a gravimetric sorption apparatus is shown in Figure 5.1.

The apparatus consists of a constant temperature diffusion chamber, which con-tains the sample, suspended from a quartz spring or (Cahn) electrobalance. Thechamber is first evacuated to a very low pressure to remove all solute from thesample, and then the diffusant is introduced at a fixed pressure. Thesorption/diffusion process is followed by recording the sorbed mass (or moisturecontent) versus time.


VB

Figure 5.1 Diagram of a gravimetric diffusion apparatus. B, balance; P, pressure;RH, constant relative humidity; T, temperature; V, vacuum.

Moisture sorption kinetics can also be measured by enclosing the sample ina constant humidity chamber at atmospheric pressure, and removing it quickly forweighing. In adsorption or desorption of water vapor, the diffusion chamber ismaintained at a fixed constant pressure, using either pure water at a specifictemperature, or constant relative humidity solutions, i.e. saturated salt solutions orsulphuric acid of fixed density (Spiess and Wolf, 1983).

The sorption process is described by the unsteady state diffusion (Pick)equation, which for one-dimensional diffusion reduces to Eq. (5-1):

dt ~ dz Bz(5-7)

The initial and boundary conditions require that the initial concentration ofthe sample is constant, the surfaces of the sample are kept constant, and theamount of diffusant (solute) is a negligible fraction of the whole. Under theseconditions, which can be achieved readily in a well-designed sorption experiment,the diffusion equation for a plane film, assuming a constant diffusivity D yieldsthe solution (Vieth, 1991)

112 Chapters

i f ~ 2 Z_i /^ . 1\2 ~ " ~ i " | r2 I > /Me

where A/ and Me are, respectively, the amounts of diffusant sorbed after time t (s)and infinity (equilibrium), D is the diffusivity (m2/s), and L is the thickness of thefilm (m). In gas sorption measurements, the pressure ratio (P/PJ can be used in-stead of the mass ratio (M/MJ, assuming that the gas law applies.

In determinations of moisture diffusivity, the mass ratio (M/MJ is equal tothe ratio of moisture contents, Y = (X-Xe)/(X0-X<i}, which in high-temperature dry-ing reduces to (X/X0), since Xe->0.

For small diffusivities and short times, Eq. (5-8) may be approximated bythe simplified equation (Crank, 1975; Vieth, 1991)

(5-9)M xL'

Thus, a plot of (M/Mg) versus ft1'2) for the first period of sorption yields anearly straight line with a slope Qf4(D/7zL2)''2, from which the diffusivity D can beestimated (Vieth, 1991). Alternatively, the sorption data can be plotted as (MM^versus ft/L2)1'2, and the diffusivity D be determined from the simplified equation(Crank, 1975; Saravacos, 1995)

[ -4 ] (5-10)D = 0.049

where (t/L2)i,2 is the "half-equilibrium time" (HE), corresponding to (M/Mg) = 0.5,i.e. 50% sorption, andZ is the thickness of the film.

Figure 5.2 shows a typical plot of adsorption of water vapor in a solid foodsample.


1.0M/Me

0.5

0.0HE (Dt/I/)2U/2

Figure 5.2 Adsorption of water vapor in a solid food sample. HE, half-equilibriumtime.

In food materials, exhibiting hysteresis, different D values of water vaporare obtained in adsorption and desorption measurements, a mean value may bemore representative of the diffusion process (Fish, 1958; Saravacos, 1967). Mois-ture equilibrium values Xe, at a given temperature, for various food materials canbe obtained from the isotherms, if available, of the specific material, from pub-lished compilations (Iglesias and Chirife, 1982; Wolf et al., 1985) or from empiri-cal equations (Saravacos, 1995).

When diffusivity changes with concentration C, which is the usual case withthe diffusivity of water in food materials, the D(C) can be estimated by repeatedmeasurements of D over various ranges of concentration (Crank, 1975). If thecritical dimension of the sample, e.g. the thickness L, changes during the sorptionmeasurement, a mean value can be used for the estimation of diffusivity(Saravacos, 1967).

2. Chromatographic MethodsGas chromatography methods, widely used in analytical chemistry, have

been proposed for the determination of sorption and diffusion properties of solidmaterials (Vieth, 1991). Of particular interest to foods is inverse gas phase chro-matography (IGPC), which can be used in the study of interaction of polymericmaterials with probes (Gilbert, 1984).

In IGPC, a polymeric material may be the stationary phase in a gas chroma-tography (GC) column, and a known probe (solute) is introduced as a mobilephase. A carrier gas transports the probe through the column where it interactswith the polymeric material, which may be coated on a GC support, e.g. diatoma-

114 Chapters

ceous earth. The chromatogram recording reveals physicochemical properties ofthe polymeric material, such as glass transition points, sorption isotherms, andthermodynamic parameters.

The specific retention volume V°, defined as the net retention volume perunit weight of polymeric material at 0°C, is given by the equation

V° = (273/T)¥a(\/Ws) (5-11)

where Vn is the net retention volume, Ws is the weight of the polymeric material,and T is the temperature (K).

The partition coefficient Kp, defined as the ratio between the probe concen-tration in the polymeric material and in the mobile phase, is given by the equation

Kf=(7°psT)/273 (5-12)

where ps is the density of the polymeric material.A technique of determining D of solutes in polymers, using IGPC, is de-

scribed by Pawlisch and Laurence (1987).Frontal Analysis (FA) may also be used in studying the transport mecha-

nisms between the mobile and stationary phases in a chromatographic column(Vieth, 1991). A mixture of carrier gas and probe vapor is forced through the col-umn at constant flow rate, and the amount adsorbed is obtained from the resultingbreakthrough curve. Solute transfer between mobile and stationary phases can beestimated by ideal chromatographic models, particularly the non-equilibrium lin-ear (LNE) model.

B. Permeability Methods

The permeability methods are convenient for estimation of mass diffusivityand are applied extensively to samples of polymer materials, which can be pre-pared in the form of thin films of homogeneous microstructure. They are difficultto apply to solid food materials, which are usually heterogeneous with holes andcracks. Permeability can be used to protective food coatings, which behave like_polymer films during the measurement (Krochta et al., 1994). The permeability P(kg / m s Pa) of a diffusant in a film is given by the equation:

P =——-—— (5-13)(AP/Az)


where J is the mass flux (kg/m2s), AP is the pressure drop (Pa), and Az is thethickness (m). _

The mass diffusivity D can be estimated from the permeability P, using thefollowing equation:

~ (5-14)

3where S is the solubility (kg/m Pa) of the diffusant (solute) in the substrate.The solubility S in a gas (vapor)/solid or gas (vapor)/liquid system is defined

by the equation:

S = CIP (5-15)

where C is the concentration of the solute in the liquid or solid substrate and P isthe partial pressure of the solute in the gas (vapor) phase.

Equation (5-15) is a form of the Henry's law, the solubility S being the in-verse of the Henry's constant (S = 1/H). The constant S is also related to the di-mensionless partition coefficient, defined as the ratio of concentrations of the sol-ute in the two contacting phases.

The solubility S of water in food materials can be estimated from the slope(dX/da) of the moisture sorption isotherm at a given temperature, either analyti-cally (empirical sorption equation), or graphically. The moisture content X (kgmoisture / kg dry matter) at a given water activity should be converted to moistureconcentration C (kg/m3), using the density of the dry matter, approximately ps -1500 kg/m3. The water activity a should be converted to partial pressure of waterP (Pa), i.e. P = aPm where P0 (Pa) is the vapor pressure of water at the given tem-perature. Thus, the slope of the isotherm is converted to (dC/dP), which has theunits of the solubility S (kg/m3Pa).

/. Time-Lag MethodThe diffusivity of a gas or vapor D and its solubility S in a polymeric film

can be estimated simultaneously from permeability measurements, using the time-lag method (Crank and Park, 1968; Crank, 1975; Vieth, 1991).

The unsteady and steady-state permeation of a solute into a polymeric filmis usually measured in a time-lag diffusion cell, shown diagrammatically in Figure5.3:

116 Chapter 5

~m wwwIAP

T

Figure 5.3 Principle of time-lag permeability/diffusion measurement. F, film; G,gas flow; S, perforated support; T, constant temperature; AP, pressure drop.

An elaborate experimental set-up is required, in which the polymeric film isfirst evacuated to a high vacuum, and a gas stream is passed through, measuringthe accumulation of the solute in the film as function of time. After a transientperiod, a steady-state permeation rate is established, which yields the permeabilityof the film P , according to Eq. (5-13). The accumulated amount of the solute inthe film Q is plotted as a function of time t, from which the diffusivity can be es-timated.

Under the specified boundary and initial conditions (initially gas-free film,equilibrium at the inlet gas/polymer interface, and zero concentration of gas at theoutflow face), and assuming a constant diffusivity D, the Pick equation yields thefollowing solution:

6D-exp - (5-16)

where Q is the accumulated amount of penetrant (solute), passing through the filmafter time t, Cj is the solute concentration in the gas upstream (high pressure), andL is the film thickness.

After a short transient period, a steady state is established, and Eq. (5-16) isreduced to:

6D(5-17)


Figure 5.4 Estimation of the time lag (TL) from a plot of accumulated diffusant Qversus time t.

A plot of (Q, t), after a short initial period, yields a straight line with an in-tercept (TL, time lag) on the t axis, given by Eq. (5-18), from which the diffusivityD can be estimated.

TL--1L6D

(5-18)

The time required to reach full steady-state diffusion has been found empiri-cally to be equal to 3(TL).

Figure 5.4 shows a typical permeability plot (Q, t) for estimation of the timelag.

2. Unsteady-State MethodThe solubility 5 and diffusivity D of gases and vapors in polymers can be

determined also by unsteady-state measurements, using specialized sorption appa-ratus (Crank, 1975; Vieth, 1991).

The polymeric material, usually a thin film of known thickness, is placedinto a closed sorption chamber, which is pressurized quickly to a known initialpressure. The pressure in the chamber drops gradually, as the gas sorbs and dif-fuses into the polymer film until equilibrium is established.

The solubility of the gas S in the polymer is estimated from a material bal-ance in the closed system, resulting in the equation:

118 Chapters

Fv 273(Pi-PlS = ——— ———- (5-19)r , T( P.

where Vv is the void volume, Vp is the volume of the polymer, T is the temperature(K), and Pk Pe are the initial and equilibrium gas pressures, respectively.

A similar expression for the solubility S can be obtained with desorptionmeasurements. The polymer sample is first completely degassed, a fixed gas pres-sure Pt is applied, the chamber is evacuated by rapid pump down to a lower pres-sure, and the chamber is left to equilibrate to a final pressure Pe.

The desorption of a gas from a polymer film by pump down of a closedchamber is described by the Pick law. Assuming that the required boundary condi-tions are met, and that diffusivity D is constant, the solution of the diffusion equa-tion yields the same equations of sorption kinetics (5-9) and (5-10), from whichthe value of D can be calculated.

C. Distribution of Diffusant

This method is based on the unsteady-state diffusion of a component in asemi-infinite solid, the contact surface of which is maintained at constant concen-tration of the diffusant. The concentration of the component in the solid is meas-ured at various distances from the surface as a function of time, obtaining the con-centration-distance curve, from which the diffusivity can be extracted (Crank,1975; Zogzas et al., 1994b; Saravacos, 1995; Kostaropoulos et al, 1994).

A common experimental procedure is to use two long cylindrical samples ofdifferent uniform concentration, contacted in series, and let diffusion take placealong the axis, under the influence of a concentration gradient. The cylindricalsample is maintained at a constant temperature, and after a certain time is removedfrom the cylinder and sliced into small sections, which are analyzed quickly forcomponent concentration. In measurements of moisture diffusivity, the sliced sec-tions of the sample are analyzed quickly for moisture gravimetrically. The samplesmay be contained in plastic cylinders (e.g. 13 mm diameter and 100 mm long),which are sliced together with the sample during the analysis (Karathanos et al.,1991). The distance-concentration curve at a specified time is constructed by plot-ting component concentration versus distance (Figure 5.5).


C,

Figure 5.5 Moisture concentration (C) - distance (z) curves in two contacted cy-lindrical samples.

The Pick diffusion equation for a semi-infinite solid yields the followingsolution, assuming constant diffusivity (Crank, 1975):

-CCc-c.r = erfc (5-20)

where C0 is the initial concentration of the diffusant in the sample, C is the concen-tration after time t, Ce is the equilibrium concentration, and z is the distance ofpenetration. The error functions erf and erfc (= 1-erf) are given in the literature.The equilibrium concentration at the given temperature is taken from the isothermof the product or from empirical equations.

For variable diffusivity, the difrusivity D(Cj) at a specified concentration C/can be evaluated by the equation:

1 (5-21)

120 Chapters

where z is the distance from the interface of the two cylinders, and t is the time atwhich the concentration profile is determined. The integral and the gradient of z atthe specified concentration Cj can be determined by numerical or graphical eva-luation of the concentration-distance curve. Thus, the diffusivity may be evaluatedas a function of concentration by repeated use of Eq. (5-21).

Both Eqs. (5-20) and (5-21) yield mean diffusivity values for both samples ofthe material. An alternative simulation method can be used to determine two sepa-rate diffusivities D for the two samples (Karathanos et al., 1991). Equation (5-20) isapplied to both cylinders, and the calculated D values are optimized by minimizingthe sum of squares of the differences of predicted and experimental concentrations.

The diffusant concentration method is convenient for estimating the diffusiv-ity of small molecules in solids, when diffusion is relatively slow, and thus concen-tration profiles can be determined within reasonable time intervals. It has the advan-tage of minimum disruption of the material structure during measurement, and ithas been applied to estimate the diffusivities of various components in food materi-als (see Chapter 7).

Application of this method to measurements of moisture diffusivity in foodsystems presents experimental difficulties, mainly due to changes of moisture con-tent during the analytical procedure (slicing, etc). The method can be used at highertemperatures and pressures, where the usual methods are difficult to apply (Kara-thanos etal., 1991).

D. Drying Methods

The drying methods are used widely for the determination of moisture (wa-ter) diffusivity of food materials, since drying and rehydration are common foodprocesses, and water transport properties are essential in modeling, calculations andcontrol of these operations. Most of the literature data were obtained from dryingexperiments, and the measurement procedure is relatively simple. Historically, dry-ing was the first method used for the determination of moisture diffusivity in solidmaterials (Sherwood, 1931).

Drying of solids is usually divided into two stages, i.e. the constant rate andthe falling rate periods (Perry and Green, 1997). In the constant rate period, waterevaporates freely from the surface of the solid, and the drying rate is controlled bythe external conditions, i.e. air velocity, temperature, and humidity. In the fallingrate period, the main resistance to mass transfer is within the solid material, and thetransport of water from the interior to the surface of evaporation is controlled bydiffusion and other mechanisms, as discussed earlier in this chapter. Most foodmaterials have short constant rate periods and they dry entirely in the falling rateperiod.


AIR

1 1w

I 1Figure 5.6 Diagram of an experimental apparatus for determining the air-dryingrate. S, sample; W, balance; u, air velocity; T, temperature; RH, relative humidity.

Determination of moisture diffusivity during the falling rate period is basedon the application of the diffusion equation to a suitable sample (slab, cylinder orsphere) of known basic dimension, assuming that the required initial and boundaryconditions are applicable. The sample is dried at controlled temperature, air velocityand relative humidity, and the drying curve is constructed by plotting the weight(moisture content) versus time. Figure 5.6 shows the principal parts of a typical air-drying apparatus for the determination of the drying rate of a solid or semisolidsample. In simple experiments, weighing of the samples may be done quickly out-side the dryer. Elaborate measuring apparatus is available, using electronic sensorsand a PC to record temperature, humidity, and sample weight during the dryingexperiment (Marinos-Kouris and Maroulis, 1995).

High air velocities (> Im/s) are used in the drying chamber in order to mini-mize the external resistance to mass transfer from the sample surface to the airstream. For vacuum-drying measurements, the air stream is replaced by a vacuumsystem with a condenser and a pressure control system. Heat can be supplied bycontrolled microwave or infrared devices.

/. Constant DiffusivityThe diffusion equation (5-1) is solved for the three basic shapes of the solids

(slab, infinite cylinder, and sphere), assuming constant diffusivity (D), and appro-priate initial and boundary conditions (Sherwood, 1931; Crank, 1975). It is alsoassumed that the solid is drying entirely in the falling rate period. For a slab or afilm of small thickness (compared to the other two dimensions) the diffusion ofmoisture is described by the equation:

122 Chapters

„ „. = JLy —— ; —— _ v~» • v - ~ • (5.22)- - 2 p 2

where A' is the mean moisture content after time t, X0 is the initial moisture content,Xe is the equilibrium moisture content, and L is the sample thickness drying fromboth sides. If the sample is drying from one flat side only, the sample thickness L inthe diffusion equation should be substituted by (2L). The moisture content A" is ex-pressed on dry basis, i.e. kg water/kg dry matter.

For spherical samples, the diffusion equation yields:

(5-23)

where r is the sphere's radius. The diffusivity D is estimated from Eqs. (5-22) or Eq.(5-23) by an approximate solution or by a numerical method. The units of D are(m2/s), provided that time is in (s) and sample thickness in (m). Most diffusivitiesare determined based on a mean value of the thickness L or the radius r of the sam-ple during the drying process.

The first reported moisture diffusivities of food materials, obtained from dry-ing data, assumed constant diffusivity (Saravacos and Charm, 1962a). However, itwas soon realized that diffusivity was a function of concentration (moisture con-tent), evidently due to the complex structure of the food materials. In some cases,e.g. drying offish muscle, the falling rate period may consist of two distinct parts.The bimodal diffusion was evidenced in a plot of log 7 versus t, which yielded twostraight lines with two slopes, Kj > Ki, corresponding to two diffusivities, £>/ > D?(Jason, 1958; Jason and Peters, 1973).

The slope K (1/s) of the semi-log plot of the drying curve is actually the dry-ing constant, defined by the equation:

X,) (5-24)at

If the diffusion equation is applicable, the diffusivity D can be estimated fromthe drying constant K, for a slab of thickness L using the equation (Moyne et al,1987; Marinos-Kouris andMaroulis, 1995):

(5-25)7t

The drying constant K refers to specific drying conditions (temperature, air humid-ity and air velocity) and sample thickness.


2. Variable DiffusivityThe moisture diffusivity in drying food materials changes significantly with

the moisture content, due to structural changes of the material during drying, andthe changes of the drying mechanism and the water-substrate interaction. Thechanges in difrusivity are evidenced by the non-linearity of the semi-log dryingcurve (logY versus f). Although mathematical solution of the diffusion equationrequires a constant difrusivity, various approximate methods are applied to extracteffective moisture diffusivities from experimental drying curves, assuming thatwater transport by all mechanisms is caused by a concentration gradient.

E. Simplified Methods

Simplified methods are useful for estimating quickly approximate values ofvariable effective diffusivities D, before a mathematically rigorous method can beapplied. They have been used for obtaining D values from irregular drying curves,as in drying of porous solids at low moistures, where moisture is transported by acombination of different mechanisms, and mathematical modeling is difficult.

The method of drying constants divides the drying curve (fog7, t) into linearparts and estimates the drying constant (K = dY/dt) at various moisture ratios, fromwhich the effective difrusivity is calculated, using Eq. (5-25) for the correspondingsample thickness.

The method of slopes (Perry and Green, 1997; Karathanos et al., 1990;Saravacos, 1995; Uzman and Sahbaz, 2000) is essentially similar to the repeatedapplication of Eq. (5-25). The experimental drying curve (logY, t) is compared tothe theoretical diffusion curve (logY, Fo) where Fo = Dt/L2 (Figure 5.7). Theslopes of the two curves (dY/dt) and (dY/dFo),h are estimated at the same mois-ture ratio Y and the effective difrusivity is calculated from the equation

(dY/dt)D = _1———i2L (5.26)(dY/dFo),, ^ '

124 Chapter 5

logY

th

tor Fo

Figure 5.7 Comparison of experimental (exp) and theoretical (th) diffusion curves.

F. Simulation Method

The simulation method estimates the effective diffusivity by an optimisationtechnique, which fits diffusivity values to the experimental drying curve (Karatha-nos et al., 1990). The input to the computer program is the drying time, the sampleshape and dimensions, the initial and equilibrium moisture contents, and the initialguess of the mean diffusivities of the nodes used (e.g. 10).

The simulation technique gave similar diffusivity values with the method ofslopes, using drying data for granular and gelatinized starch samples.

G. Numerical Methods

Numerical methods assume that the effective diffusivity D is a known func-tion of the moisture content X, and the diffusion equation (5-1) is fitted to the ex-perimental drying data by regression analysis. In a more general approach, amathematical model is proposed that considers both heat and mass transfer, andthe diffusivity is a function of moisture content and temperature (Kiranoudis et al.,1992, 1995).

An empirical model for moisture diffusivity is the following exponentialexpression (Kiranoudis et al., 1994; Marinos-Kouris and Maroulis, 1995):

D = Do expX

(5-27)


where A' is the moisture content, Tis the temperature, and X0, T0 are adjustable con-stants.

A large number of drying data is obtained from drying experiments at fixeddrying conditions, and fitted simultaneously to the diffusion equation by a non-linear regression technique. Two iterative methods of calculation can be used, i.e.the finite differences of Crank-Nicolson (Crank, 1975) and the control volume (Pa-tankar, 1980).

A numerical method, based on the exponential function of moisture diffusiv-ity, (Eq. 5-27), was applied to the air-drying of potato and carrot. Comparison withthe simplified method of drying constants gave acceptable agreement in the lowmoisture range, where a complex transport mechanism has been evidenced (Kira-noudisetal., 1994).

Specialized mathematical models for the effective moisture diffusivity D maydescribe irregular changes during the drying process. For example, the bell-shapedcurve of (D, X) curve, observed in drying porous starch materials at low moistures(X<1) can be represented by the gamma function (Karathanos et al, 1990):

1 A -

(5-28)

where /I, j3 are constants, X is the moisture, Xz = X-XRH=O, and F(/3) is the gammafunction.

H. Regular Regime Method

The regular regime method, developed by Schoeber (1976), is based on theapplication of mass transfer principles to the experimental drying curve. Themethod involves a number of calculations, as outlined briefly below (Schoeber andThijessen, 1977; Gekas, 1992).

The drying process involves three periods: the constant rate, the penetration,and the regular regime. During the regular regime (last) period of drying, the mois-ture profile is moving toward the center of the product, depending on the diffusivityof moisture, but not on the initial moisture content of the product.

From the drying curve (X, t) the mass flux per unit external surface of thesample is calculated (dm/dt, kg/m2s), and then the characteristic flux parameter F =(dm /dt) ps (L~/2), kg2/m4s, where ps is the dry solids concentration (kg/m3) and L isthe sample thickness (m). The moisture profile in the sample is described by therelationship FY= constant, where Y is the moisture ratio, Y = (X0-X)/(X0-X^>.

A curve (F, X) is constructed, which constitutes the regular regime curve forthe given material.

126 Chapters

The gradient d(lnF)/d(lnX) is calculated as a function of moisture content Xby numerical differentiation, and correlated to the Sherwood number (Sh = kcL/D,where kc is the mass transfer coefficient, m/s). A plot of (2F/Sh) versus A'yields thereduced diffusivity Dr by numerical differentiation. Finally, the mass diffusivity Dis calculated from the reduced diffusivity, using the equation, D = Dr/ps

2.The regular regime method has found limited applications to food materials

(Singh et al, 1984; Tong and Lund, 1990; Sano and Yamamoto, 1990; Inazu andIwasaki, 2000). The calculated moisture diffusivities are, in general, very close tothe diffusivities obtained by other methods for the same food material.

I. Shrinkage Effect

Most food materials undergo significant shrinkage during the drying process,which is reflected in the calculated values of moisture diffusivity, since D is nor-mally proportional to the square of the sample thickness. Shrinkage models andexperimental data are presented in Chapter 3 (Zogzas et al., 1994a; Krokida andMaroulis, 1997). In general, shrinkage is a linear function of moisture content, andit should be determined for each material under the appropriate drying conditions.Moisture diffusivity values D, calculated for a mean sample thickness, can be con-verted to values based on the dry solids of the material Ds, using the followingequation (Fish, 1958; Crank, 1975):

Ds = [(p/ps)(l+X)]2/3D (5-29)

where p and ps are the densities of the sample at moisture content X and dry-ness, respectively. The volumetric shrinkage is assumed to be isotropic. Gekas andLamberg (1991) modified the Crank equation (5-29), assuming that shrinkage is notisotropic, and the moisture diffusivity Df follows a fractal relationship:

Dr[(p/Ps)]2/dD (5-30)

where d is the fractal exponent, e.g. for drying of blanched potato d = 1.42.


IV. MOISTURE DIFFUSIVITY IN MODEL FOOD MATERIALS

Model food systems are useful in understanding and predicting the behaviorof actual food materials in various heat and mass transport processes. Moisturetransport (diffusivity) has been found to vary widely in food materials, due mainlyto different physical structure.

Starch materials have been used in several simulations of food products,since starch is a basic structural component of various foods of plant origin, e.g.cereal and potato products. Other food biopolymers can also be used in foodsimulations, like pectin, cellulose and various gums. Starch (usually corn orpotato) has the advantage of forming rigid gels when heated, resembling solidfood products. The use of food biopolymers in measurements of moisturediffusivity can utilize the experience and advances of polymer science in the areaof physical and physicochemical properties.

A. Effect of Measurement Method

The method of measurement may have a profound effect on the value ofmoisture diffusivity of solid and semisolid foods, due to the changes in the physi-cal structure of the sample during the measurement procedure. In liquid foods, asin pure liquids and gases, the experimental or computed diffusivities are morelikely to be constant for a given set of conditions (see Chapter 2).

Drying causes significant physical changes in the food sample during meas-urement (formation of cracks, pores, puffing), strongly affecting the estimatedmass transport property (moisture diffusivity). Isothermal adsorption or desorptionof water is a milder transport process, with no major structural changes. As a re-sult, the moisture diffusivity D calculated from sorption data is usually lower thanthe D value obtained from drying data. The method of concentration distributiondoes not change the physical structure of the material during measurement, and theD values obtained are similar to those of sorption.

Although, theoretically, diffusivity should be independent of sample shapeand thickness, different results are obtained in practice when changing sampledimensions, especially in drying measurements. Thus, thicker samples yield highD values, evidently due to the formation of larger cracks, pores and channels dur-ing drying.

Figure 5.8 shows moisture diffusivities D in starch gels at 25°C, obtained bysorption kinetics. Thin films (1 mm) of potato starch gels were used, and the re-ported diffusivities were the mean values of adsorption and desorption measure-ments (Fish, 1958). The D values increased exponentially from 1 x 10"13 to 0.5 x10"10 m2/s when the moisture content^was increased from 0 to 0.2.

128 Chapters

The increase of moisture diffusivity can be explained by the plasticization ofthe polymer (starch), which facilitates the transport of water through the macromo-lecular network. The thin samples of starch gel behaved like homogeneous poly-mer materials, and the small moisture diffusivities are of the same order of magni-tude with the D values of small molecules through polymeric materials (Vieth,1991).

Temperature has a positive effect on moisture diffusivity, following the Ar-rhenius equation, in a similar way with the activated diffusion in pure liquids,which has been explained by the theory of rate processes (Eq. 2-37). The energy ofactivation for diffusion ED increases substantially as the moisture content is re-duced, e.g. ED = 20 to 41 kJ/mol, at moisture contents X= 0.2 to 0.01, respec-tively.

Higher moisture diffusivities are obtained in starch gels, using drying ratemeasurements (Saravacos and Raouzeos, 1983). The D values shown in Figure 5.9were obtained by drying slabs of starch gels 6 mm thick in an airstream at 2 m/s. Amaximum of D = 5xlO"10m2/s was observed at nearly X= 2. Incorporation of glu-cose (50% dry basis) in the starch gel reduced significantly the moisture diffusiv-ity.

The higher D values obtained from drying experiments, compared to thesorption data, are the result of significant changes in the structure of the gel duringthe drying process. Drying experiments require samples much thicker than the thinfilms of sorption measurements. Drying increases the porosity and creates cracksand channels in the sample, through which water can be transported at a faster rateas a vapor than in the isothermal sorption process (Chapter 3). The presence ofsmall, water-soluble molecules in the gel, like glucose, reduces moisture diffusiv-ity, by decreasing the porosity of the sample during drying.

A comparison of moisture diffusivities D obtained from drying and sorptionexperiments on the same starch material is shown in Figure 5.10. Granular cornstarch (high-amylose, HYLON) was used to prepare spherical samples 1 cm indiameter for the drying measurements at 60°C and air velocity 2 m/s. The samestarch material, in the form of a slab (film) 1 mm thick was used in the sorptionmeasurements (Leslie et al., 1991; Chung, 1991).

In both cases, the diffusivity-moisture content (D versus X) curve goesthrough a maximum, which is lower moisture content for the sorption data. HigherD values (nearly 3 times) were obtained by the drying method, evidently due tothe structural changes in the samples during air-drying.


0.0010.1 0.2

X (kg/kg dm)

0.3

Figure 5.8 Moisture diffusivity in potato starch gels. Sorption kinetics, 25°C.(Data from Fish, 1958.)

0 1 2 3 4

X (kg/kg dm)

Figure 5.9 Effective moisture diffusivity in drying slabs of starch gels at 40°C.S, starch; SG, starch glucose. (Data from Saravacos and Raouzeos, 1983.)

130 Chapter 5

0.4 0.6

X (kg/kg dm)

0.8

Figure 5.10 Comparison of moisture diffusivities in granular starch (AMIOCA),obtained from drying (DR) and adsorption (ADS) at 60°C. (Data from Leslie et al.,1991.)

B. Effect of Gelatinization and Extrusion

Gelatinization causes significant physicochemical and structural changes instarch materials, which considerably affects the heat and mass transport properties(Chapter 3). The moisture diffiisivity is, in general, reduced by gelatinization, dueto the disruption of the granular structure and the formation of a homogeneous gel.The effect of gelatinization on diffusivity depends primarily on the chemical andphysical composition of the starch material.

Gelatinization of high-amylose (linear macromolecules) starch (HYLON)causes a relatively small reduction of moisture diffusiviry, without changing thecharacteristic shape of the (D, X) curve of porous materials, as shown in Figure5.11. The gelatinized spherical sample developed a high porosity with severalcracks, which resulted in relatively high moisture diffusivity. A completely differ-ent effect of gelatinization on moisture diffusivity was observed in high-amylopectin (branched macromolecules) starch (AMIOCA), as shown in Figure5.12(Saravacosetal., 1989).


The moisture diffusivity D in the gelatinized starch was reduced sharply, es-pecially at low moisture contents. The values of the gelatinized starch increased asthe moisture content X was increased, resembling the (D, X) curve obtained bysorption kinetics of starch gels. The moisture diffusivity curves of Figure 5.12 canbe explained by the changes of bulk porosity s as a function of moisture content.In the gelatinized high-amylopectin gel, the porosity increased only slightly duringthe drying process, contrary to the sharp increase in the granular (non-gelatinized)sample (Figure 3.7).

Extrusion cooking of starch material at high temperatures and relatively lowmoisture contents, produces highly porous products with high moisture diffusivity.Figure 5.13 shows typical moisture diffusivities in high-amylopectin extrudedstarch, obtained from drying measurements of extruded cylindrical samples.

0.2 0.4 0.6

X (kg/kg dm)

Figure 5.11 Effective moisture diffusivity in drying spherical samples of granular(GR) and gelatinized (GEL) starch (HYLON) at 60°C. (Data from Saravacos et al.,1989.)

132 Chapter 5

0.2 0.4 0.6

X (kg/kg dm)

Figure 5.12 Effective moisture diffusivity in drying spherical samples of granular(GR) and gelatinized (GEL) starch (AMIOCA) at 60°C. (Data from Saravacos etal., 1989.)

50

40

N 30

o

I 20a

10

0.2

EX

GEL

0.4 0.6

X (kg/kg dm)

Figure 5.13 Effective moisture diffusivity in drying of extruded (EX) and gelati-nized (GEL) starch (AMIOCA) at 60°C. (Data from Marousis et al.,1991.)


C. Effect of Sugars

Incorporation of small water-soluble molecules into the starch materials re-duces, in general, the moisture diffusivity. The effect of sugars is more pro-nounced in granular (non-gelatinized) than in gelatinized starch materials. Figure5.14 shows typical effects of sugars on he moisture diffusivity D in high-amylose(HYLON) starch, obtained from drying experiments using spherical samples 1 cmin diameter at 60°C. The D values decreased sharply when water-soluble dextrinwas incorporated in the samples. Smaller effects were observed with glucose andsucrose (Marousis et al., 1989).

The molecular weight of the water-soluble carbohydrate seems to be propor-tional to the reduction of moisture diffusivity. It is presumed that the water-solublemolecules precipitate in the starch matrix during drying, reducing significantly theporosity of the drying sample. This explanation is supported by the observed lowporosities of osmotically (using sugar) dehydrated fruits, compared to the un-treated materials (Chapter 3).

The effect of sugars on the moisture diffusivity in gelatinized starches islower than in granular materials (Figure 5.15). Low molecular weight sugars, likeglucose, appear to have a stronger effect because they are more mobile in the gelstructure, precipitate in pores created during drying, and reduce moisture diffusiv-ity (see also Figure 5.9). On the other hand, high-molecular weight carbohydrates,like dextrin, are relatively immobile, and they behave like starch, creating highporosity during drying, and increasing moisture diffusivity.

Dissolved salts, such as sodium chloride, reduce the moisture diffusivitystrongly in hydrated granular starches, but slightly in the gelatinized materials(Uzman and Sahbaz, 2000). Sodium chloride, which is mobile in granular starch,may concentrate and precipitate near the surface of the sample, causing a "case-hardening" effect, and reducing substantially the effective moisture diffusivity. Itshould be noted that the diffusivity of sodium chloride in water is about 12xlO"10

m2/s (Table 2.4).

134 Chapter 5

0.2 0.4 0.6

X (kg/kg dm)

Figure 5.14 Effect of sugars on moisture diffusivity in drying granular starch (HY-LON) at 60°C. S, starch; SG, starch/glucose; SD, starch/dextrin. (Data from Ma-rousisetal., 1989.)

0.4 0.6

X (kg/kg dm)

0.8

Figure 5.15 Effect of sugar on moisture diffusivity in drying gelatinized starch(HYLON) at 60°C. S, starch; SG, starch/glucose; SD starch/dextrose. (Data fromMarousis et al, 1989.)


D. Effect of Proteins and Lipids

Incorporation of proteins in starch materials may substantially reduce themoisture diffusivity D due to physicochemical interactions. Figure 5.16 shows thereduction of D in granular amylopectin (branched macromolecule) by the additionof 25% (by dry weight) gluten (a wheat protein). The D values were estimatedfrom drying measurements on spherical samples at 60°C. A similar effect wasobserved with the addition of 25% lysozyme (Marousis, 1989).

Addition of proteins to amylose (linear macromolecule) reduced the D val-ues to a lesser degree than amylopectin, presumably due to the weaker pro-tein/starch interaction, compared to the stronger interaction of proteins with thebranched macromolecules of amylopectin.

Proteins reduce moisture diffusivity in gelatinized starches in a similar man-ner with the granular materials, suggesting a molecular type of interaction. Bycontrast, addition of sugars reduces the D values of granular starches only, due tosignificant reduction of porosity (a physical process), but there is little effect in thegelatinized materials.

Incorporation of lipids in granular starches may reduce the moisture diffu-sivity D, but there is little effect in the gelatinized materials (Papantonis, 1991).Figure 5.17 shows a significant reduction in D by the addition of 10% vegetableoil in granular amylopectin starch during drying of spherical samples at 60°C.However, there is no significant change in D of gelatinized starches by the addi-tion of lipids, suggesting a rather physical than chemical interaction.

Measurements of moisture diffusivity in model food systems containingstarch, proteins and lipids, demonstrate the importance of physical and physico-chemical interactions on the transport of water in actual food materials. Proteinsappear to have a strong physicochemical effect, reducing the mobility of watermolecules. Lipids may reduce moisture diffusivity by physical obstruction of wa-ter transport, e.g. by forming hydrophobic films in the starch network.

136 Chapter 5

0.2 0.4 0.6

X (kg/kg dm)

Figure 5.16 Effect of gluten (SP) on the moisture diffusivity (D) of granular amy-lopectin starch (S). Drying of spherical samples at 60°C. (Data from Marousis,1989.)

0.2 0,4 0.6

X (kg/kg dm)

0.8

Figure 5.17 Effect of vegetable oil (SO) on the moisture diffusivity (D) of granularamylopectin starch (S). Drying of spherical samples at 60°C. (Data from Papan-tonis, 1991.)


E. Effect of Inert ParticlesInert particles, i.e. solid particles not interacting with water or biopolymers,

may support the physical and mechanical structure of starch materials during thedrying process. Shrinkage of the samples may be prevented, and the moisture dif-fusivity may be increased during drying (Leslie et al, 1991).

Figure 5.18 shows that the moisture diffusivity of hydrated granular starch(HYLON) increases significantly, when silica particles (25% by weight, dry basis)are incorporated in the spherical samples. A similar effect was observed whencarbon black was incorporated in the starch samples. The bulk porosity s of thedried starches, containing inert particles, increased significantly, e.g. for hydratedHYLON starch e increased from 0.45 to 0.52 (silica) and 0.55 (carbon black), andfor AMIOCA starch the increase was from 0.45 to 0.50 (silica) and 0.57 (carbonblack).

0.4 0.6X (kg/kg dm)

0.8

Figure 5.18 Effect of inert silica particles (SI) on moisture diffusivity of granularstarch S (HYLON). Air drying at 60°C. (Data from Leslie et al., 1991.)

138 Chapters

F. Effect of Pressure

Pressure has a significant effect on the moisture diffusivity of porous mate-rials, such as granular starches. Pressure can be applied in the form of mechanicalcompression or gas (air) pressure in a closed vessel.

As shown in Figure 5.19, the effective moisture diffusivity D in granularstarch at moisture content X = 0.5 decreased from about 10xlO~10to 3x10"'° m2/s,when the mechanical pressure was increased from 1 to 40 bar (Marousis et al.,1990). The reduction of D is related directly to the reduction of porosity by theapplied mechanical pressure. The effect of pressure on the gelatinized starch mate-rials was relatively smaller, corresponding to smaller changes of porosity.

Mechanical pressure reduces the porosity of granular starches, especially athigh moisture contents, when the starch granules can be deformed more easilythan the dry particles. Figure 5.20 shows that the porosity of granular starch atmoisture content X = 0.5 is reduced from about 0.50 to less than 0.10 when themechanical pressure is increased from 1 to 40 bar.

Air pressure applied in a closed vessel to granular starch (HYLON) reducedthe moisture diffusivity D in a similar manner with mechanical pressure. Thus, Ddecreased from about 10xlO"'°to 2xlO"10 m2/s, when the air pressure was increasedfrom 1 to 40 bar (Figure 5.21). The values of D were determined by the moisturedistribution method (Karathanos et al., 1991).

The effect of gas (air) pressure on diffusivity in porous materials is related tothe inverse pressure P - diffusivity D relationship in gas systems at constant tem-perature, according to the simplified equation,

PD = constant (5-31)

Higher moisture diffusivities D are expected in porous materials at pressure belowatmospheric (vacuum), an important characteristic of vacuum and freeze-drying.Thus, the moisture diffusivity in freeze-dried starch gels increased from aboutO.lxlO'10 m2/s to 10x10"'° m2/s, when the pressure was reduced from 1 bar to be-low 1 mbar (Saravacos and Stinchfield, 1965). The diffusion data were obtainedby the sorption kinetics method.


Figure 5.19 Effect of mechanical pressure on the moisture diffusivity of granularstarch (HYLON) at moisture X = 0.5. Drying at 60°C. (Data from Marousis et al.,1990.)

0.6

0.4

0.2

0 10 20 30 40 50

P (bar)

Figure 5.20 Effect of mechanical pressure on the porosity e of granular starch (HY-LON) at moistureX= 0.5. (Data from Marousis et al., 1990.)

140 Chapter 5

P (bar)

Figure 5.21 Effect of air pressure on the moisture diffusivity of granular starch(AMIOCA) at moisture X= 0.4. Moisture distribution method at 60°C. (Data fromKarathanos et al, 1991.)

G. Effect of Porosity

The bulk porosity s of solid materials, estimated from measurements of bulkand solids density (Figure 3.5), is the major parameter affecting mass diffusivity.Temperature and moisture content are also important, but their effect dependsstrongly on the structure of the material. Regression analysis of several experimen-tal data on granular and gelatinized starch materials has yielded the followingequation (Marousis et al., 1991):

(5-32)

Figure 5.22 shows that the effective moisture diffusivity increases sharply above e= 0.40. The pore size distribution of dried starch materials is discussed in Chapter3.

The high porosity, developed in the drying of granular starch, is visualizedby the formation of flow channels, through which water (liquid and vapor) is


transported to the drying surface. Radial channels are formed in drying sphericalsamples. Irregular cracks are formed in drying gelatinized starch (Figure 3.8).

The pore shape has a significant effect on the heat and mass transport prop-erties of solids. Freeze-drying experiments have shown that moisture diffusivity ishigher in samples with long than small pores (Figure 5.23; Saravacos, 1965). TheCMC gel slab, which dried faster, had a fibrous structure with long pores orientedalong the diffusion path, while the starch gel sample had small spherical pores,distributed evenly (Figure 3.9).

H. Effect of Temperature

The effect of temperature on moisture diffusivity D depends strongly on thephysical structure of the solid material. In porous materials, where vapor diffusionmay be controlling, D is proportional to the (3/2) power of temperature, accordingto the fundamental transport equation (2-28). In nonporous gels, where liquid dif-fusion may predominate, D may be expressed by the Arrhenius equation (2-37):

D = Aexp(-ED/RT) (5-33)

The energy of activation for diffusion ED, estimated from diffusivity data at vari-ous temperatures, is a good indication of the type of prevailing diffusion mecha-nism in the material. In general, low ED values indicate a vapor diffusion, whilehigh values suggest liquid (activated) diffusion (Table 5.2). Higher ED values areexpected at low moisture contents, due to the stronger water-substrate interaction.

Table 5.2 Typical Energies of Activation for Diffusion of Water in Starch Materi-als__________________________________________Material____________________Activation energy, kJ / mol____Granular starch 17.0Granular starch/sugar 33.5Gelatinized starch 43.4Gelatinized starch/sugar 51.4Granular starch/sodium chloride 61.0

142 Chapter 5

0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 5.22 Effect of porosity on moisture diffusivity of starch materials. (Datafrom Marousis et al, 1991.)

0.01

0.001

Figure 5.23 Freeze drying rates of model food gels. S, potato starch; CMC, car-boxy methyl cellulose. Y = (X-Xe)/(Xo-Xe). (Data from Saravacos, 1965.)


I. Drying Mechanisms

Figure 5.24 shows a typical moisture diffusivity-moisture content (D, X)curve for a porous food material, e.g. granular, extruded, or freeze-dried foodproduct. At high moistures (A), e.g. X > 0.5, liquid diffusion and capillary flowmay predominate (relatively low D and low porosity). At intermediate moistures(B), water may be transported by capillary flow and vapor diffusion, increasingsharply the D value. In this region, pores, cracks, and channels facilitate the trans-port of water. At very low moistures (C), strongly bound water is desorbed fromthe biopolymer solids, reducing sharply the moisture diffusivity.

The shape of the (D, X) curve of Figure 5.24 is characteristic of mass trans-port properties of capillary and porous solids at low concentrations (Toei, 1983). Asimilar curve is obtained for the change of thermal diffusivity of porous foods inthe low moisture region (Figure 8.7), suggesting a heat and mass transport anal-ogy.

40

30

20

10

0.2 0.4 O . f

X (kg/kg dm)

0.8

Figure 5.24 Prevailing mechanisms of water transport in porous food materials. A,liquid diffusion and capillary flow; B, vapor diffusion; C, desorption of sorbed wa-ter.

144 Chapters

V. WATER TRANSPORT IN FOODS

The transport of water in food materials is an important physical process inthe processing, storage, quality and utilization of food products. Of particular im-portance is the transport of water within the mass of solid and semisolid foods,since external transfer to and from the surrounding environment can be understoodand analyzed more readily, based on the principles and applications of interphaseheat and mass transfer (Chapter 9).

Data and conclusions from model food systems are useful in obtaining ap-proximate values of mass transport properties of broad classes of foods, e.g. po-rous, gelatinized, and sugar-containing products. However, the structure of realfoods of both plant and animal origin is more complex, containing cells, cellularmaterials, membranes, fibers, etc making difficult the exact physical modeling foranalysis of the transport processes. It becomes, therefore, necessary to obtain ex-perimental data on transport properties of broad classes of foods and individualfood products.

In this section, water transport properties (mainly moisture diffusivity) ofvarious foods are discussed on the basis of physical and physicochemical struc-ture, and some characteristic values are given. A more detailed analysis of theliterature data on moisture diffusivity of foods is presented in Chapter 6. Reviewsof moisture diffusivity of foods were presented by Bruin and Luyben (1980),Chirife (1983), Saravacos (1995, 1998), Zogzas et al. (1996) and Mittal (1999).

Lists of moisture diffusivities are given in some food engineering books(Gekas, 1992; Okos et al., 1992), and in a food properties database (Singh, 1993).A more detailed database was developed in the European Union (European Coop-erative Project FAIR "DOPPOF").

A. Mechanisms of Water Transport

Most of the data on moisture diffusivity have been obtained from drying ex-periments, since mass transfer within the food material is the rate-controlling resis-tance. Two drying periods are usually observed, the constant rate and the fallingrate. The constant rate period is controlled by external conditions of heat and masstransfer, and the interphase transport coefficients are discussed in Chapter 9.

In most foods, drying takes place mainly in the falling rate period, and inter-nal mass transport becomes very important. Water is transported within the foodmaterials by a combination of several mechanisms, depending on the physicalstructure of the product and the external drying conditions. The prevalent mecha-nisms are molecular diffusion (liquid and vapor), capillary flow, and hydrody-namic flow. Other mechanisms may be also involved, such as Knudsen diffusion,surface diffusion, and thermal diffusion (Soret effect).


B. Effective Moisture Diffusivity

Molecular diffusion, described by the Pick equation (5-1), is used widely forthe estimation of the effective moisture diffusivity D of foods, although water maybe transported by mechanisms other than diffusion. It is assumed that the drivingforce for all water transport is the concentration gradient (dC/dz) or the moisturecontent gradient (dX/dz). The methods of determination of D are discussed in Sec-tion III of this chapter.

The drying curve (logY, t), obtained from drying experiments of specifiedsamples under controlled conditions, provides useful information on the mecha-nism of moisture transport, and it is utilized for the determination of the effectivemoisture diffusivity. The drying ratio is defined as Y = (X-Xg)/(X0-X' J, where X0,X, and X e, respectively, are the moisture contents (kg/kg dm) at the beginning,after time (t), and at equilibrium.

Semi-logarithmic plots of logY versus / may result in straight lines, an indi-cation that the diffusion equation may be applied for the treatment of the dryingdata. Low (negative) slopes d(logY)/dt of the drying curve indicate external resis-tance to mass transfer, while high (negative) slopes characterize internal resistance(Mulet, 1994).

The applicability of the diffusion equation to the transport of water duringdrying of foods can be also tested by the following simple techniques: a) Increas-ing the air velocity should not have a significant effect on the drying rate; b) theBiot number Bi = kcL/D (where kc is the mass transfer coefficient, m/s, L is thesample thickness, m, and D the diffusivity, m2/s) should be very high, e.g. Bi >1000; and c) the drying time in the falling rate period should be proportional to thesquare of the sample thickness (Saravacos and Charm, 1962).

A constant slope of the drying curve indicates a constant moisture diffusiv-ity D, which can be estimated by analytical or numerical techniques. However, inmost foods, the slope is not constant, suggesting that D is a function of the mois-ture content. In some cases, there are two straight lines with decreasing slopes,from which two D values can be estimated (bimodal diffusion). An example ofbimodal diffusion is shown in Figure 5.25, and it refers to the air-drying of codfishfillet (Jason, 1958). The fish slab had a thickness of 15 mm, and it was dried at35°C and air velocity of 3.7 m/s. Two effective moisture diffusivities were esti-mated from the two slopes (drying constants), D, = 3.4xlO'10 and D2 = 0.8x10"'°m2/s.

The broken drying curve shows that, after some drying period, the moisturediffusivity decreases significantly, evidently due to shrinkage of the fish muscle,without pore formation. The increased resistance to moisture transfer at lowermoisture contents is shown from the increase of the energy of activation for diffu-sion ED from 30 to 37 kJ/mol.

146 Chapter 5

0.01

0.001

Figure 5.25 Drying curve (bimodal diffusion) of codfish fillet at 35°C. Slab thick-ness 15mm. Y=(X-Xe)/(Xo-Xe). (Data from Jason, 1958.)

C. Water Transport in Cellular Foods

Both plant and animal foods consist basically of cellular tissues of variouscomponents, including water, biopolymers, sugars, salts, membranes, cell walls,fibers, etc. In food processing, the heat and mass transport processes consider non-living food materials, i.e. the physiological processes of the living cells are ne-glected. The physiological processes are normally disrupted by heating, freezing,and dehydration of the food products.

Mass transport in cellular foods can be analyzed thermodynamically by thechemical potential approach, instead of the usual concentration gradient of Pick'sequation (Rotstein, 1987; Gekas, 1992; Doulia et al., 2000). The chemical poten-tial of water (kJ/mol) is related to the water activity aw by the equation:

=RTln(aw) (5-34)

The effective moisture diffusivity D is related to the chemical potential gra-dient by the equation:


D = [Kfa)/pJ (dn/cK) (5-35)

where K(ju) is the effective mass conductivity based on the chemical potentialgradient, with units kg kmol/s kJ, and ps is the solids density, kg/m3. The chemi-cal potential can be estimated from the water activity, using Eq. (5-34). Predictionof water activity is discussed by Rahman (1995).

The effective mass conductivity K(p) is the summation of the mass conduc-tivities of all cellular components in all phases, including gas diffusivities andpermeabilities. The required values of porosity, tortuosity, and sorption equilibriaare estimated from the physical properties of the food system.

Although the chemical potential approach is thermodynamically sounderthan the concentration gradient (diffusion) method, limited applications have beenreported in the literature, evidently due to the involved calculations and the lack ofreliable data on structural properties of the food material, such as porosity andtortuosity.

D. Water Transport in Osmotic Dehydration

In osmotic dehydration, water is removed from the food material, due to anosmotic pressure gradient between the contacting osmotic solution and the prod-uct. Two transport processes take place simultaneously, i.e. water loss (WL) fromthe product and sugar gain (SG) into the product. Osmotic solutions used in prac-tice are sugar solutions, e.g. 65°Brix glucose or sucrose, and salt (sodium chloride)solutions.

The transport of water and solute in osmotic dehydration at atmosphericpressure is usually modeled as a diffusion process, and the Pick equation is appliedto samples of known dimensions at specified boundary conditions (Lazarides etal., 1997). Typical effective moisture diffusivity in apple tissues in contact with asucrose solution at 25°C is D = 5xlO"10m2/s. The diffusivity of sucrose in the samesystem is lower as expected: D = Ixl0"10m2/s. The diffusivity of sodium chloridein osmotic treatments is close to the diffusivity of salt in water at the same tem-perature and concentration, e.g. D = lOxlO"10 m2/s (Table 2.4).

In vacuum osmotic dehydration (VOD), mass transport (water and solute) ismainly by hydrodynamic flow than by molecular diffusion (Fito, 1994). Hydrody-namic flow, due to mechanical pressure gradients, is facilitated by the existence ofintercellular pores in the food material, e.g. apple tissues, which can be evacuatedand filled with the osmotic solution by mechanical flow. The osmotic process canbe accelerated by pulsed VOD.

Water loss WL and sugar gain SG during osmotic dehydration can be mod-eled by the following empirical kinetic models (Panagiotou et al., 1998a; 1998b):

148 Chapters

WL / WLe = 1 - exp(- KWL t) (5-36)

SG /SGe = 1 - exp(- KSG t) (5-37)

where WL, and SGe are the equilibrium water loss and sugar gain, respectively (attime t -> ac), and KWL, KSG are the corresponding constants. The rate constants KWLand KSG are related to empirical parameters of sample size, solute concentrationand molecular weight, temperature, and flow conditions (speed of agitation).

The empirical models of Eqs. (5-36) and (5-37) were applied to the osmoticdehydration of apple, banana, and kiwi. Glucose is a better osmotic agent thansucrose, because of its lower molecular weight and the higher mobility. The os-motic dehydration of fruit with sugars has significant effects on the moisture diffu-sivity in the dried product. Determination of moisture diffusivity D from air-drying data of osmotically treated apples yielded lower D values, evidently due tothe effect of dissolved sugars (Karathanos et al., 1995). The diffusivity - moisturecontent curve (Figure 5.26) is similar to the curves of sugar containing starch ma-terials (Figure 5.14). The dissolved sugar molecules precipitate in the pores of thefruit during drying, reducing significantly the porosity of the dried product (Chap-ter 3). Cylindrical apple samples 10 mm diameter were dried at 55°C and 2 m/s airvelocity.

Sorption kinetics data show a similar effect of sugars on the moisture diffu-sivity D in osmotically treated apples (Bakalis et al., 1994). Figure 5.27 shows thechanges in D during adsorption of water vapor in dried apple samples 8 mm di-ameter in humidified air. The D values obtained from sorption experiments weresignificantly lower than the data obtained by the drying method, in a similar man-ner with the data obtained on model food materials (see Section IV).


(SS

1 2 3 4 5

X (kg/kg dm)

Figure 5.26 Effective moisture diffusivity in air-drying (55°C) of osmo-treatedapples. A, untreated apple; AS, osmo-treated apple. (Data from Karathanos et al.,1995.)

0.1 0,2 0.3

X (kg/kg dm)

0.4

Figure 5.27 Effective moisture diffusivity of water vapor in osmo-treated driedapples, obtained from sorption measurements. A, untreated sample; AS, osmo-treated sample. (Data from Bakalis et al., 1994.)

150 Chapters

E. Effect of Physical Structure

As shown in model food systems, the physical and physicochemical struc-ture of foods has a decisive effect on the mechanism of water transport and theeffective moisture diffusivity. Figure 5.28 shows the effect of drying method onthe moisture diffusivity of apple, using the sorption kinetics technique (Saravacos,1967).

Apple slices were dehydrated by air-drying, explosion-puff drying, andfreeze-drying, resulting in products of entirely different structure. Thin slices ofthe dried materials, 1-2 mm thick, were used in adsorption and desorption meas-urements in a vacuum sorption apparatus at 30°C and increasing water vapor pres-sures. The obtained low values of moisture diffusivity are characteristic of sorp-tion measurements on thin samples, resembling the sorption behavior of polymer(starch) films (Fish, 1958).

It is evident that the moisture diffusivity D in the freeze-dried material ismuch higher than in the puff-dried and air-dried samples, over the entire moisturecontent range. The (D-X) curve for the puff-dried sample resembles the curves ofgranular starches with an intermediate maximum, a characteristic behavior of po-rous food materials. By contrast, the moisture diffusivity in the air-dried samplewas the lowest, and it increased steadily at higher moisture contents. This behaviorcharacterizes nonporous food materials and polymers (starches), in which diffu-sion of water is facilitated by moisture adsorption. Similar effects on moisturediffusivity are observed with potato samples, dehydrated by the three differentdrying methods described previously for the apple (Figure 5.29). A significantincrease of moisture diffusivity in oil containing foods is obtained by solventextraction of the oil.

Figure 5.30 shows the moisture diffusivities D of untreated and defattedsoybeans, using the sorption technique. The measurements were made on thinslices, 1 mm thick, in a vacuum apparatus at 30°C. The D values of untreated sam-ples increases steadily from O.OlxlO"10 to 0.03xlO~10m2/s as the moisture content isincreased from 0.2 to 0.12 kg/kg dm, a characteristic of sorption in low porositybiopolymers. By contrast, a higher D value was obtained in the defatted sample(0.055xlO"10m2/s), which remained constant over the same moisture content range.

The effect of shrinkage and porosity on the moisture diffusivity of fruits andvegetables is similar to the effect on model foods (Zogzas et al., 1994a; KrokidaandMaroulis, 1997).


o.oi0.1 0.2

X (kg/kg dm)

0.3 0.4

Figure 5.28 Effect of drying method on the moisture diffusivity in apple. Sorptionkinetics on thin samples, 30°C. AD, air-dried; PD, puff-dried; FD, freeze-dried.(Data from Saravacos, 1967.)

o.oi0.00 0.05 0.10

X (kg/kg dm)

0.15 0.20

Figure 5.29 Effect of drying method on the moisture diffusivity in potato. Sorp-tion kinetics on thin samples, 30°C. AD, air-dried; PD, puff-dried; FD, freeze-dried. (Data from Saravacos, 1967.)

152 Chapter 5

(SEo

0.010.00 0.10

X (kg/kg dm)0.20

Figure 5.30 Moisture diffusivity of full-fat (F) and defatted (DF) soybeans. Sorp-tion kinetics on thin slices at 30°C. (Data from Saravacos, 1967.)

The decrease of moisture diffusivity in freeze-dried, and in porous foodsgenerally, at high moisture contents, is directly related to a collapse of the physicalstructure (Karel and Flink, 1983).

F. Effect of Physical/Chemical Treatments

Various physical treatments during the processing of foods may change theirphysical structure, resulting in changes of moisture diffusivity. As discussed inSection III, mechanical compression, gelatinization, coating, and mixing with hy-drophilic food components may reduce substantially the moisture diffusivity, dueprimarily to the reduction of porosity. On the other hand, puffing (extrusion, ex-plosion puffing), vacuum and microwave treatment may increase porosity and,therefore, moisture diffusivity.

Starch gelatinization reduces considerably moisture diffusivity in cookingand boiling starch-based foods in water, creating an outside layer of slowly mov-ing water. The water demand of the inner core may be considered as the drivingforce for water transport inwards. The moisture diffusivity in such food systemscan be measured by the NMR technique (Takeuchi et al., 1997; Gomi et al., 1998).


Microwave or dielectric treatment of foods can substantially increase themoisture diffusivity during drying at atmospheric pressure or in vacuum. Micro-wave energy is absorbed by the water inside the food product, resulting in fastevaporation, and creating a porous structure in the dried material. Pretreatment offruits and vegetables with microwave energy and subsequent normal air-dehydration improves the drying rate.

Short-time microwave pretreatment of grapes significantly increases thedrying rate of grapes, due to an increase of the moisture permeability of the grapeskin. The pretreatment can be applied to the sun drying of grapes (raisins), whichis normally very slow, and a long time is required for drying (Kostaropoulos andSaravacos, 1995). Short-time (0.5-3.0 min) microwave pretreatment of starch gelsand apples considerably increased the drying rate in subsequent air-drying. Themean effective moisture diffusivity, estimated from the drying curves, increasedfrom 9xlO'10to 18xlO"'°m2/s in the starch gels and from 6xlO''°to 12xlO'10m2/s inthe apples (Saravacos et al., 1997).

The skin of some fruits, such as grapes, reduces substantially mass transferduring drying, due to the presence of waxes and other hydrophobic components,that have low moisture permeability and diffusivity. Chemical pretreatments canincrease water permeability by dissolving the waxes and breaking down the skinstructure.

Alkali dips in solutions of sodium hydroxide, containing surfactants (surfaceactive agents), are applied to grapes before sun drying, substantially reducing thedrying time. Ethyl oleate, an edible surfactant, considerably increases the dryingrate by acting on the skin and the grape berry during the drying process. Figure5.31 shows the effect of addition of 2% ethyl oleate to a solution of 0.5% NaOH,which was used as pretreatment dip in the air-drying of seedless grapes (Saravacosand Raouzeos, 1986). A similar effect of alkali dips on the drying of grapes wasreported by Masi and Riva (1988). The mean moisture diffusivity in the alkali-treated grapes was increased by the addition of ethyl oleate from 0.3x10"10 to1.0xlO-'°m2/s.

Low concentrations of surfactants sharply reduce the surface tension of wa-ter. Addition of surfactants can increase the drying rate of porous food materialsby increasing the wetting of the drying surfaces during the early stages of drying.This effect has been observed in the drying of porous fruits, such as apples(Saravacos and Charm, 1962b). However, surfactants have little effect on the dry-ing rate of nonporous food materials, such as blanched potato.

Similar effects of ethyl oleate were observed in the air-drying of granular(porous) starch (Saravacos et al., 1988). On the other hand, the surfactant treat-ment had no effect on the drying rate of gelatinized starch, a low-porosity modelfood material (see Figure 5-32).

154 Chapter 5

Figure 5.31 Effect of alkali/surfactant dip on the air-drying of grapes at 60°C. CL,control (0.5% NaOH); TR, treated (0.5% NaOH +2% ethyl oleate).Y = (X-Xe)/(Xo-Xe). (Data from Saravacos and Raouzeos, 1986.)

Figure 5.32 Effect of ethyl oleate on the drying rate at 60°C of granular starch(HYLON). CL, control; TR, treated 0.2% ethyl oleate. 7 = (X-Xe)f(Xo-Xe). (Datafrom Saravacos et al., 1988.)


G. Characteristic Moisture Diffusivities of Foods

Characteristic values of moisture diffusivity D of various food products areuseful as a first approximation of this important transport property in the designand evaluation of food processes and food storage/quality changes. The widevariation of he literature data on D is due to the complex physical and chemicalstructure of foods and the different methods of measurement and estimation used.A unified analysis of the literature data on D is presented in Chapter 6.

Table 5.3 gives some typical D values for important classes of foods, basedon the physical structure and porosity. The effect of structure of foods is similar tothe effect of structure on the D of model food materials, as discussed in SectionIV. Tables 5.4 to 5.8 give typical D values of various food products, selected fromthe literature.

Table 5.3 Effect of Physical Structure on the Moisture Diffusivity of Food Prod-uctsFood Structure Porosity Activation energy Diffusivity______________%_______kJ/mol_________xlQ-'V/s

Highly porous 90 10 50Freeze-driedPuff-driedFibrous

Porous 50 20 20GranularVacuum-dried

Low porosity 20 30 5Compressed

Gelatinized 10 45 1

Starch/protein foods 10 50 0.1

Starch/lipid foods 10 60 0.01

Glassy-state foods 0 >60 0.0001

156 Chapters

Table 5.4 Typical Moisture Diffusivities of Cereal Grains and Rice at 30°C(typical activation energy ED = 40 kJ/mol)Food material Moisture Diffusivity___________________kg/kg dm______xlQ-'V/sCorn kernel 0.20 0.40Corn pericarp 0.20 0.01Wheat kernel 0.20 0.50Rice 0.20 0.40

Table 5.5 Typical Moisture Diffusivities of Baked Products and Pasta at 30°C(typical activation energy ED = 40 kJ/mol)Food material

DoughBreadCookiePastaPuffed pasta

Table 5.6 Typical(typical activationFood material

PotatoCarrotPeasOnionSoybeans

Moisturekg/kg dm

0.400.300.150.150.15

Moisture Diffusivities of Vegetableenergy ED = 45 kJ/mol)

Moisturekg/kg dm

0.300.300.100.100.20

DiffusivityxlO'10m2/s

5.02.00.50.31.2

Products at 30°C

DiffusivityxlO-'W/s

5.02.03.00.50.8


Table 5.7 Typical Moisture Diffusivities of Fruit Products at 30°C(typical activation energy ED = 60 kJ/mol)Food material Moisture Diffusivity_____________________kg/kg dm______xlQ-'°m2/sApple 0.50 2.0Apricots 0.40 1.0Bananas 0.50 2.0Raisins 0.40 1.5

Table 5.8 Typical Moisture Diffusivities of Meat and Fish at 30°C(typical activation energy ED = 35 kJ/mol)Food material

Minced beefPork sausageCodfishHerringMackerelHulibut

Moisturekg/kg dm

0.600.200.500.500.400.40

DiffusivityxlO-'W/s

1.00.52.00.80.50.3

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McCarthy, M.J., Perez, E., Ozilgan, M. 1991. Model for Transient Moisture Pro-files of a Drying Apple Slab Using Data Obtained with Magnetic ResonanceImaging. Biotechnol Progr 7:540-543.

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Moisture Diffusivity Compilation ofLiterature Data for Food Materials

I. INTRODUCTION

There is a wide variation of the reported experimental data of moisture dif-fusivity of solid food materials, making difficult their utilization in food processand food quality applications.

The variation of moisture diffusivity in model and real foods is discussed inChapter 5. The physical structure of solid foods plays a decisive role not only onthe absolute value of moisture diffusivity, but also on the effect of moisture con-tent and temperature on this transport property. Porosity data, obtained frommeasurements of bulk and particle densities (Chapter 3) at various moisture con-tents, provide useful data on the type of water transport (liquid or vapor diffusion)and the approximate value of moisture diffusivity.

In this chapter, the moisture diffusivity in food materials is approached froma statistical standpoint. Literature data are treated by regression analysis, using anempirical mathematical model. Recently published values of moisture diffusivityin various foods were retrieved from the literature and were classified and ana-lyzed statistically to reveal the influence of material moisture content and tempera-ture. Empirical models relating moisture diffusivity to material moisture contentand temperature were fitted to all examined data for each material. The data werescreened carefully, using residual analysis techniques. A promising model wasproposed based on an Arrhenius-type effect of temperature, which uses a parallelstructural model to take into account the effect of material moisture content.

163

164 Chapters

II. DATA COMPILATION

Moisture diffusivity data on foods in the literature are scarce because of theeffect of the following factors: (a) diverse experimental methods, (b) differentmethods of analysis used, (c) Variation in composition of the material, and (d)variation of the structure of the material.

Literature data for moisture diffusivity in foods materials were selected andpresented in the reviews of Bruin and Luyben (1980), Chirife (1983), Gekas(1992), Marinos-Kouris and Maroulis (1995), Zogzas et al. (1996), Mittal (1999),and Doulia et al. (2000). In addition to these reviews, an exhaustive literaturesearch was made in international food engineering and food science journals inrecent years, as follows (Panagiotou et al., 2001):

• Drying Technology, 1983-1999• Journal of Food Science, 1981-1999• International Journal for Food Science and Technology, 1988-1999• Journal of Food Engineering, 1983-1999• Trans. of the ASAE, 1975-1999

A total of 175 papers were retrieved from the above journals according thedistribution presented in Figure 6.1. The accumulation of the papers versus thepublishing time is presented in Figure 6.2. The search resulted in 1558 data con-cerning the moisture diffusivity in food materials. The 1558 data retrieved fromthe above search, plus 16 data from Bruin and Luyben (1980), 58 data fromChirife (1983) and 141 data from Gekas (1992) were organized into a databaseand analyzed. A total number of 1773 data was obtained. These data are plottedversus moisture and temperature in Figures 6.3 and 6.4, respectively. These fig-ures show a good picture concerning the range of variation of diffusivity, moistureand temperature values. More than 95% of the data are in the ranges:

• Diffusivity IxlO'1 2-IxlO"6m2 /s• Moisture 0.01 - 15.0 kg/kg db• Temperature 10-200°C

It should be noted that the lowest possible moisture diffusivity is that inglassy food materials, about IxlO"14 m2/s, and the highest is the diffusion of watervapor in air at atmospheric pressure, IxlO"5 m2/s (Table 2.4). Higher diffusivitiesof water vapor can be obtained in vacuum systems. The self diffusivity of water isnear Ixl0"9m2/s.

Moisture Diffusivity Data Compilation 165

J. FoodEngineering

DryingTechnology

J. of FoodScience

Trans of theASAE

Int. J, FoodScience and

Techn.

Figure 6.1 Number of papers on moisture diffusivity data in food materials pub-lished in food engineering and food science journals during recent years.

1000 -1—

1975 1985 1995 2005

Year

Figure 6.2 Accumulation of published papers on moisture diffusivity data for foodmaterials versus time.

166 Chapter 6

l.E-03

l.E-06 - =

•5 l.E-09

l.E-12

l.E-150.01 0.1 1 10

Moisture (kg/kg db)100

Figure 6.3 Moisture diffusivity data for all foods at various moistures.

l.E-03

l.E-1510 100

Temperature (°C)1000

Figure 6.4 Moisture diffusiviry data for all foods at various temperatures.


The histogram in Figure 6.5 shows the distribution of the moisture diffusiv-ity values retrieved from the literature. The results obtained are presented in detailin Tables 6.1-6.3. More than 100 food materials are incorporated in these tables.They are classified into 11 food categories. Table 6.1 shows the related publica-tions for every food material. Table 6.2 summarizes the average literature valuefor each material along with the corresponding average values of correspondingmoisture and temperature. Table 6.3 presents the range of variation of moisturediffusivity for each material along with the corresponding ranges of moisture andtemperature.

1000

1)•cIa•MaO

BsZ

100

10

l.E-15 l.E-12 l.E-09 l.E-06

Moisture Diffusivity Values (mVs)l.E-03

Figure 6,5 Histogram of observed values of moisture diffusivity in food materials.

168 Chapter 6

Table 6.1 Literature for Moisture Diffusivity Data in Food Materials:References and Number of Data Retrieved

Material

Baked Products

Biscuit

Bread

Cookie

Crackers

Dough

Cereal Products

Barley

Brown rice

Reference

Balasubrahmanyam and Datta, 1993

Zhouetal., 1994

Gekas, 1992Lomauroetal, 1985

Kirn and Okos, 1999

Kamthanos et al., 1995Zanonietal, 1994

Fasina et a!., 1998Miketinac et al, 1992

Engelsetal, 1986Gekas, 1992Hendrickx et al, 1988

Data

33

4411211

1818844

499

1679

36973

Lu and Siebenmorgen, 1992Steffe and Singh, 1980


Table 6.1 ContinuedMaterial Reference

CornBakker-Arkema et a!., 1987Galan-Domingo and Martinez-Vera, 1996Gekas, 1992Harosetal, 1995Jumah and Mujumdar, 1996Martinet- Vera eta!., 1 995Mouradetal, 1996Muthukumarappan and Gunasekaran, 1994a, b, cParti and Dugmanics, 1990Patil, 1988Shivhare et al, 1992Syariefetal., 1987Tolabaetal, 1989Tolabaetal, 1990Verma and Prasad, 1999Waltonetal, 1988Zahedetai, 1995

MaltLopezetal, 1997

Milled riceZhangetal, 1984

Paddy riceBase et al, 1987

Parboiled brown riceIgathinathane and Chattopadhyay, 1999a, b

Parboiled paddy riceIgathinathane and Chattopadhyay, 1999a, b

Parboiled riceChandra and Singh, 1984Igathinathane and Chattopadhyay, 1999a, b

PastaLitchfield and Okos, 1992Meat et al, 1996Waananen and Okos, 1996Xiongetal, 1991

Data

168633965

1740

88

109847

2054422336666

1266

51114

306

170 Chapter 6

Table 6.1 ContinuedMaterial

Rice

Rough rice

Wheat

Wild rice

Reference

Engelsetal, 1986Galan-Domingo and Martinez-Vera, 1996Gekas, 1992Hendrickx et al, 1986Hendrickx and Tobback, 1987Hendrickx etal, 1988Steffe and Singh, 1980Zahedetal, 1995

Bakker-Arkema et al, 1987Chungu and Jindal, 1993EceandCihan, 1993Gekas, 1992Lague andJenkins, 1991Lu and Siebenmorgen, 1992Sarkeretal, 1994Steffe and Singh, 1980Suarezetal, 1982Tang and Sokhansanj, 1993

Bakker-Arkema et al, 1987Bruin and Luyben, 1980Chirife, 1983Devahastin et al, 1998Gekas, 1992Giner and Calvelo, 1987Gineretal, 1996Gong etal, 1997Igathinathane and Chattopadhyay, 1997Jar os et al, 1992Jayas et al, 1991Lomauro et al, 1985Sapru and Labuza, 1996Suarezetal, 1982Sun and Woods, 1994Zahedetal, 1995

Gekas, 1992

Data

5663

15368

105

78366

1468

21914

58323344444791115333



Dairy

Cheese

Dry milk

Milk

Skim milk

Fish

Catfish

Cod

Dogfish

Fish meal

Haddock

Halibut

Herring

Mackerel

Reference

Change! al, 1998Desobry and Hardy, 1994Luna and Chavez, 1992

Gekas, 1992Lomauroetal, 1985

Straatsmaetal., 1999

Ferrari et al, 1989Kerkhof, 1994

Chirife, 1983

Balaban and Pigott, 1988Chirife, 1983

Bruin and Luyben, 1980Chirife, 1983

Alvarez and Shene, 1996Blasco and Alvarez, 1999

Chirife, 1983

Chirife, 1983

Chirife, 1983

Chirife, 1983

Data

32

925221166

1596

63

11642413

158722227711

172 Chapter 6


Shark

Squid

Swordfish

Whiting

Fruits

Apple

Apricot

Avocado

Banana

Blueberries

Coconut

Reference

Park, 1998

Teixeira and Tobinaga, 1995Teixeira and Tobinaga, 1998

Chirife, 1983

Chirife, 1983

Bruin and Luyben, 1980Chirife, 1983Fuscoetai, 1991Gekas, 1992Lazarides et al., 1997Lomauroetal, 1985Nieto et al, 1998Simaletal, 1997Simaletal, 1998a, b

Abdelhaq and Labuza, 1987Vagenas and Marinos-Kouris, 1991

Chirife, 1983Gekas, 1992

Garciaetal, 1988Gekas, 1992Johnson et al, 1998Mauro and Menegalli, 1995Rastogietal, 1997Son/cat el al., 1996Waliszewskietal, 1997

Nsonzi and Ramas\vamy, 1998Ramaswamy and Nsonzi, 1998

Gekas, 1992

Data

12126154433

268

64362

1413

16

118

1028

1165

555639

12119

29111866



Grapes

Mulberry

Peach

Pineapple

Raisins

Legumes

Broad bean

Fababean

Lentil

Navy beans

Meat

Beef

Reference

Alvarez and Legues, 1986Gekas, 1992Mahmutogluetal, 1996Simaletal, 1996Vagenasetal, 1990

Mas/can andGogus, 1998

Gekas, 1992

Azuaraetal, 1992Beristainetal, 1990Rastogi and Niranjan, 1998

Gekas, 1992Karathanos et al, 1995Lomauro et al, 1985Sapru and Labuza, 1996

Ptaszniketal, 1990

Hsu, 1983a, b

Tang and Sokhansanj, 1993

Radajewski et al, 1992

Gekas, 1992Huang and Mittal, 1995

Data

52184

16869911

20695

111811

29

7722

121288

53

422

174 Chapters


Beef carcass

Broiled

Bull

Chicken

Ground beef

Heifer

Pepperoni

Sausage

Turkey

Model foods

Albumin-flour-bran

Amioca

Cellulose-oil-water

Corn starch

Flour

Reference

Mallikarjunan and Mittal, 1994

McLendon and Gillespie, 1978

Gekas, 1992

Ngadi et al, 1997

Gekas, 1992Hallstrom, 1990Lomauroetai, 1985

Gekas, 1992


Dincer and Yildiz, 1996

Chirife, 1983Gekas, 1992

Strumitto et al, 1996

Karathanos et al., 1990Kostaropoulos and Saravacos, 1997Marousisetal, 1989Seowetal., 1999Vagenas and Karathanos, 1993

Chirife, 1983

Gekas, 1992

Gekas, 1992Lomauroetai., 1985

Data

33333366

191

1717753211211

202

272759164

201091122312


Table 6.1 Continued_________________________________Material Reference Data

Glucose-starch 7Gekas, 1992 1

Gluten-starch 6Xiong et a!., 1991 6

Hylon-7 59Karathanosetal, 1990 9Kostaropoulos and Saravacos, 1997 4Marousisetal, 1989 18Seawetal.,1999 11Tsukadaetal, 1991 9Vagenas and Karathanos, 1993 8

Polyacrylamide gel 6Roques et a!., 1994 6

Potato starch 4Gekas, 1992 4

Rice starch 23GOBI; e/ a/., 1998 3Takeuchi et al, 1997 20

Starch 5Geto, 7P92 5

Nuts_______________________________79

Almond 2Beviaetai, 1999 2

Hazelnuts 12Lopezetal, 1998 12

Peanut pods 64Chinnan and Young, 1977a, b 64

Peanuts 1SuarezetaL, 1982 1

Other_______________________________45

Canola 24Thakoretal., 1999 24

Chocolate 4Biquet and Labuza, 1988 4

176 Chapter 6


Coffee

Egg

Sunflower seeds

Toria

Vegetables

Beet

Broccoli

Carrot

Cassava

Garlic

Okra

Reference

Gekas, 1992

Kincal, 1987

Rovedo et al, 1993

Raoetal, 1992

Chirife, 1983

Sanjuanetal, 1999Simaletal, 1998a, b

Cordova-Quirozet al, 1996Gekas, 1992Kiranoudis et al, 1992Kiranoudis et al., 1993Kompany et al., 1993Mabrouk and Belghith, 1995Markov/ski, 1997Muletetal., 1987Mulet et al, 1989Mulet, 1994Rastogi and Raghavarao, 1997Stapley et al, 1995

Chavez-Mendez et al, 1998Fuscoetal, 1991

Madambaetal, 1996Pezzutti and Crapiste, 1997Pinagaetal, 1984Vazquezetal, 1999

Gogus and Maskan, 1999

Data

33664444

470

11

221012

106149

15108

127

126

12101028

22565666



Onion

Paprika

Pea

Pepper

Pigeon pea

Potato

Reference

Baroni and Hubinger, 1998Kiranoudis et al., 1992Lewicki et al, 1998LopezetaL, 1995

Gekas, 1992

Medeiros and Sereno, 1994

Carbonelletal, 1986Kiranoudis et al, 1992

Shepherd and Bhardwaj, 1988

AfzalandAbe, 1998Bonetal, 1997Bruin and Luyben, 1980Chirife, 1983Costa and Oliveira, 1999Fuscoetal, 1991Gekas and Lamberg, 1991Gekas, 1992Kiranoudis et al, 1992Lazarides and Mavroudis, 1996Lazaridesetal, 1997Magee and Wilkinson, 1992Maroulis et al., 1995McLaughlin and Magee, 1999McMinn and Magee, 1996Mishkinetai, 1984Mulet, 1994Pinthus et al, 1997Rice and Gamble, 1989Rovedo and Viollaz, 1998Rovedoetal, 1995Rubnov and Saguy, 1997Yusheng and Poulsen, 1988Zhouetal, 1994

Data

31999433

99

145955

16512647424

129618

158

1298

123334

121

178 Chapter 6


Soya meal

Soybean

Sugar beet

Tapioca

Tomato

Turnip

Yam

Reference

Alvarez and Blasco, 1999Alvarez and Shene, 1996

Barrozo et al, 1998Deshpande et al., 1994Gekas, 1992Hsu, 1983a, bMisra and Young, 1980Oliveira and Haghighi, 1998Suarezetal, 1982

Bruin and Luyben, 1980Chirife, 1983Fuscoetal, 1991


Dincer and Dost, 1995Hawladeretal., 1991Karatas and Esin, 1994

Lomauro et al., 1985Moreiraetal, 1993

Hawladeretal, 1999

Data

18126

1931184117232413

16385

101922


Table 6.2 Moisture Diffusivity of Foods Versus Moisture and Temperature:Average Values of Available Data

Material

Baked ProductsBiscuit

fondant coatedBread

-Cookie

oatmealCrackers

-Dough

-

Cereal ProductsBarleykernel

Brown rice-

branendosperm

kerneltesta

Corn-

dentendosperm

flintgerm

grainshard endosperm

kernelpericarpsemident

shelledsoft endosperm

without pericarp

Diffusivity(m2/s)

5.26E-09

5.00E-08

3.99E-12

6.08E-10

4.89E-10

1.85E-10

2.64E-102.59E-116.68E-114.36E-114.46E-11

3.27E-073.40E-101.08E-103.20E-111.31E-106.04E-101.72E-114.49E-111.09E-114.54E-111.01E-102.96E-114.06E-13

Moisture(kg/kg db)

0.12

0.67

0.18

0.08

0.48

0.23

0.190.220.240.240.24

0.340.280.200.500.140.280.090.350.190.500.190.060.21

Temperature(°C)

85

80

25

63

122

49

4045404040

41604055354833493855713333

No. ofData

33441122

181888

49299

36189333

168311533

11284

25133

2048

180 Chapter 6


Malt-

Milled rice-

Paddy ricegrains

Parboiled brown ricebran

Parboiled paddy ricehusk

Parboiled riceendospermlong grain

short grainPasta

-dense

porousRice

-cooking

endospermgrains

hullkernel

testaRough rice

-bran

endospermgrains

hullhusk

kernel

Diffusivity Moisture Temperature(m2/s) (kg/kg db) (°C)

8.73E-08

1.31E-10

1.53E-11

3.52E-11

1.01E-IO

3.21E-106.05E-111.37E-10

1.94E-114.76E-111.21E-10

4.49E-1 14.13E-099.94E-111.05E-101.20E-111.21E-104.43E-11

3.71E-098.68E-121.73E-101.30E-111.06E-112.56E-111.62E-08

0.45

0.32

0.11

0.33

0.30

0.520.500.50

0.160.130.12

0.26

0.230.26

0.23

0.190.150.150.150.230.150.28

50

60

50

75

75

756060

638168

371303961454040

49434741454560

No. ofData

4422336666

12633

5121181256132

2132

123

7835

86197

12



Wheat-

branendosperm

flakesgrains

hardkernel

shreddedsoft

Wild ricebroken

unprocessedwhole

DairyCheese

-Dry milk

nonfatMilk

powderSkim milk

-

FishCatfish

-Cod

-muscle

Dogfish-

Fish meal-

Haddock-

muscle

Diffusivity(m2/s)

1.45E-101.73E-101.91E-108.33E-143.69E-112.02E-096.54E-115.53E-121.51E-10

7.00E-134.00E-132.00E-13

2.02E-08

2.12E-11

6.58E-10

1.36E-10

8.00E-1 1

2.78E-103.40E-10

1.48E-10

7.97E-10

6.00E-113.30E-10

Moisture(kg/kg db)

0.230.350.500.500.150.170.160.110.15

0.58

0.12

0.30

0.56

3.00

0.40

Temperature(°C)

493030256620632550

252020

9

25

40

46

30

4230

30

118

3030

No. ofData

5826221

102

11133111

32992266

1515

631165144

1515211

182 Chapter 6


Halibut-

muscleHerring

-Mackerel

-SharkmuscleSquid

mantleSwordflsh

-salted

Whiting-

muscle

FruitsApple

-Apricot

-Avocado

-Banana

-plantain

ripeBlueberries

-Coconut

-Grapes

-red

seedlessMulberry

-


5.80E-112.50E-10

6.53E-11

3.50E-11

1.80E-10 1.31

8.91E-11 1.50

3.45E-102.95E-10

4.80E-111.76E-10

6.64E-10 2.80

1.39E-07 2.88

6.35E-10

1.4 IE-09 1.636.51E-10 0.901.43E-09 3.00

2.12E-10 1.75

9.77E-10 0.60

1.37E-101.79E-102.03E-10 1.49

1.18E-09 1.50

3030

30

30

30

34

4848

3028

47

53

50

535560

45

83

536060

70

No. ofData

2117711

121266422312

268646410101111554933

292966

52182

3299



Peach-

Pineapple-

Raisins-

LegumesBroad bean

seedsFababean

-Lentil

cotyledonshilum

seedcoatNavy beans

•

MeatBeef

-meatball

rawBeef carcass

bonefat

muscleBroiled

wasteBull

-Ground beef

-heat treated

raw

Diffusivity(mVs)

8.00E-12

1.47E-09

1.67E-10

6.53E-07

1.78E-07

2.25E-117.22E-091.57E-12

4.56E-08

5.56E-103.20E-10l.OOE-11

5.48E-123.07E-115.83E-10

9.81E-06

7.40E-1 1

3.03E-111.48E-108.61E-11

Moisture(kg/kg db)

4.50

0.37

0.26

0.75

0.130.150.18

0.28

1.40

0.30

0.76

0.160.801.13

Temperature(°C)

30

40

37

30

30

404040

50

3014030

21

27

255143

No. ofData

11

20201111

297722

1233688

53412131113333

19289

184 Chapter 6


Heiferheat treated

rawPepperoni

-sausageSausage

-Turkey

-

Model foodsAlbumin-flour-bran

mixture!mixture 2mixturesAmioca

-gel

hydratedCellulose-oil-water

-Corn starch

-Flour

-Glucoose-starch

-Gluten-starch

gelatinizedungelatinized

Hylon-7-

gelhydrated

Polyacrylamide gel-

Potato starch.

Diffusivity(m2/s)

1.69E-108.33E-11

5.20E-115.33E-11

1.31E-07

8.00E-15

1.45E-091.0 IE-097.70E-10

2.26E-098.20E-101.90E-09

3.10E-09

2.25E-10

2.26E-1 1

2.27E-10

3.33E-112.67E-11

2.09E-092.06E-092.27E-09

1.52E-10

6.91E-12

Moisture(kg/kg db)

1.001.00

0.19

0.32

0.04

0.430.490.49

0.360.330.33

0.12

0.60

0.140.14

0.300.440.33

0.90

25.28

Temperature(°C)

5143

1212

180

22

929292

746060

68

30

25

39

7474

636060

40

25

No. ofData

7435231122

20227999

5949

5511223377633

5948

656644



Rice starchfull heatednonheated

Starch-

NutsAlmond

-Hazelnuts

shelledunshelled

Peanut podshull

kernelPeanuts

-

OtherCanolaembryo

kernelChocolate

darkCoffeeextract

Eggfresh

incubatedSunflower seeds

hullkernelToriaseeds


9.75E-102.1 IE-09

4.23E-10

2.32E-12

4.03E-096.16E-09

4.69E-117.28E-11

4.00E-11

3.58E-095.09E-09

1.03E-13

1.08E-10

1.44E-111.61E-11

4.40E-101.20E-10

9.80E-11

1.631.50

0.60

0.05

0.150.15

0.600.60

0.10

0.020.02

2.00

1.000.80

0.070.07

0.10

5057

42

281

5555

3535

50

8080

20

50

3636

4545

65

No. ofData

23131055

7922

1266

64323211

45241212443363342244

186 Chapter 6


VegetablesBeet

-Broccoli

stemsCarrot

-Cassava

rootsGarlic

-Okra

-Onion

-Paprika

-Pea

-Pepper

-redpo-wderPigeon pea

kernelPotato

-restructured product

tissueSoya meal

-Soybean

-grains

Sugar beetroots

Tapiocaroots

Diffusivity Moisture Temperature(mVs) (kg/kg db) (°C)

1.50E-09

1.29E-09

2.05E-09

6.30E-10

1.74E-10

2.24E-09

1.0 IE-09

2.17E-10

2.74E-10

6.22E-092.09E-10

5.07E-11

1.32E-092.02E-091.67E-09

1.16E-08

9.0 IE-081.12E-09

6.59E-10

6.00E-10

8.80

4.60

0.63

0.80

2,00

1.65

0.97

3.700.06

0.20

3.17

1.85

0.10

0.560.60

2.60

1.05

65

62

53

67

50

70

64

48

48

7049

70

5810565

162

4748

57

78

No. ofData

46711

2222

1061061010222266

31313399

149555

165148

161

1818161327744



Tomato-

concentrate dropletsTurnip

-Yam

_

Diffusivity(m2/s)

7.57E-101.87E-09

1.64E-09

1.27E-09

Moisture(kg/kg db)

10.000.50

6.33

0.10

Temperature(°C)

6477

57

45

No. ofData

161 1

5101022

188 Chapter 6

Table 6.3 Moisture Diffusivity of Foods Versus Moisture and Temperature:Variation Range of Available Data

Material min

Diffusivity(m2/s)

max

Moisture

min(db)max

Temperature

min(°C)max

Baked ProductsBiscuit

fondant coatedBread

-Cookie

oatmealCrackers

-Dough

-

Cereal ProductsBarleykernel

Brown Rice-

branendosperm

kerneltesta

Corn-

dentendosperm

flintgerm

grainshard endosperm

kernelpericarpsemident

shelledsoft endosperm

without pericarp

3.97E-I24.06E-094.06E-095.00E-085.00E-083.97E-123.97E-121.40E-111.40E-111.30E-101.30E-10

8.33E-141.31E-111.31E-111.81E-121.81E-121.48E-114.36E-112.00E-111.25E-119.72E-142.33E-112.39E-113.61E-111.90E-115.28E-125.24E-111.29E-111.11E-119.72E-142.50E-112.16E-112.12E-112.56E-13

5.00E-086.29E-096.29E-095.00E-085.00E-084.00E-124.00E-121.81E-091.81E-09l.OOE-09l.OOE-09

4.04E-066.52E-106.52E-103.94E-093.94E-094.07E-119.64E-117.25E-119.31E-114.04E-064.04E-061.48E-092.01E-104.50E-118.42E-106.1 IE-092.06E-1 11.14E-107.37E-116.80E-112.23E-103.72E-115.50E-13

0.030.030.030.670.670.180.180.030.030.200.20

0.020.100.100.160.160.210.240.240.240.060.100.100.100.500.100.100.070.120.100.500.190.060.19

0.670.390.390.670.670.180.180.140.140.600.60

0.560.270.270.250.250.240.240.240.240.560.500.500.300.500.300.400.100.560.300.500.190.070.23

1577778080252540401515

530301212353030301010304045251025252545382525

2039191808025259090

203203

1507070

12012055505050

12012090406540

12040906065

1044040


Table 6.3 Continued

Material

Malt-

Milled Rice-

Paddy Ricegrains

Parboiled Brown Ricebran

Parboiled Paddy Ricehusk

Parboiled Riceendospermlong grain

short grainPasta

-dense

porousRice

-cooking

endospermgrains

hullkernel

testaRough Rice

-bran

endospermgrains

hullhusk

kernel

min

1.11E-081.1 IE-088.33E-118.33E-112.28E-122.28E-121.15E-111.15E-113.43E-113.43E-112.38E-112.12E-102.62E-112.38E-111.55E-121.55E-129.40E-122.43E-113.30E-123.30E-121.92E-094.40E-116.67E-114.00E-125.19E-111.20E-117.56E-137.56E-134.03E-129.86E-111.30E-114.08E-121.11E-114.75E-12

Diffusivity(m2/s)

max

2.14E-072.14E-071.78E-101.78E-103.44E-113.44E-116.26E-116.26E-111.80E-101.80E-104.84E-104.84E-101.02E-102.98E-103.42E-104.84E-1 11.06E-103.42E-106.33E-091.17E-106.33E-091.98E-101.48E-102.00E-112.28E-109.30E-111.68E-072.64E-082.21E-112.20E-101.30E-113.59E-114.41E-111.68E-07

Moisture(db)

min max

0.450.450.130.130.110.110.300.300.280.280.500.510.500.500.020.020.050.040.100.10

0.200.22

0.23

0.050.100.150.150.150.220.150.05

0.450.450.500.500.110.110.360.360.330.330.530.530.500.500.320.320.230.210.350.35

0.250.30

0.24

0.500.330.150.150.150.240.150.50

Temperature(°C)

min max

2020606040405050505040504040404040408

20110

8613530301212253041353038

808060606060

1001001001001001008080

12285

12210515055

1505561555050

120120606041556082

190 Chapter 6

Table 6.3 Continued

Material

Wheat-

branendosperm

flakesgrains

hardkernel

shreddedsoft

Wild Ricebroken

unprocessedwhole

DairyCheese

-Dry Milk

nonfatMilk

powderSkim Milk

FishCatfish

-Cod

-muscle

min

8.33E-146.07E-121.68E-101.91E-108.33E-141.50E-113.30E-IO1.15E-115.53E-122.98E-112.00E-137.00E-134.00E-132.00E-13

2.10E-115.60E-115.60E-112.10E-112.10E-113.50E-113.50E-112.51E-112.51E-11

1.30E-118.00E-118.00E-118.10E-118.10E-113.40E-10

Diffusivity(m2/s)

max

3.70E-095.30E-101.78E-101.92E-108.33E-147.94E-1 13.70E-091.43E-105.53E-123.19E-107.00E-137.00E-134.00E-132.00E-13

9.00E-089.00E-089.00E-082.13E-112.13E-111.83E-091.83E-092.56E-102.56E-10

1.89E-098.00E-118.00E-115.13E-105.13E-103.40E-10

Moisture

min

0.100.170.350.500.500.100.130.100.110.15

0.120.350.350.120.120.200.200.200.20

0.33

3.003.00

(db)max

0.500.300.350.500.500.200.200.300.110.15

0.800.800.800.120.120.400.400.800.80

3.00

3.003.00

Temperature

min

55

303025402040253020252020

000

252510103030

203030303030

(°C)max

8685303025862080257025252020

701313252570707070

1703030606030


Table 6.3 Continued

Material

Dogfish-

Fish Meal-

Haddock-

muscleHalibut

-muscle

Herring-

Mackerel-

SharkmuscleSquid

mantleSwordfish

-salted

Whiting-

muscle

FruitsApple

-Apricot

-Avocado

-Banana

-plantain

ripe

min

8.30E-118.30E-111.95E-111.95E-116.00E-116.00E-1 13.30E-105.80E-115.80E-112.50E-101.30E-111.30E-113.50E-113.50E-118.70E-118.70E-118.30E-118.30E-112.60E-103.00E-102.60E-104.80E-114.80E-118.20E-11

4.00E-134.00E-124.00E-12l.OOE-11l .OOE-111.10E-101.10E-101.60E-101.60E-103.16E-105.50E-10

Diffusivity(m2/s)

max

2.20E-102.20E-101.89E-091.89E-093.30E-106.00E-113.30E-102.50E-105.80E-112.50E-101.90E-101.90E-103.50E-113.50E-112.85E-102.85E-101.09E-101.09E-103.90E-103.90E-103.30E-102.70E-104.80E-112.70E-10

6.10E-076.40E-096.40E-096.10E-076.10E-071.80E-091.80E-093.40E-093.40E-091.15E-092.66E-09

Moisture

min

0.330.33

1.181 . 1 80.500.50

0.000.000.000.500.50

0.250.250.903.00

(db)max

0.550.55

1.421.422.502.50

8.708.708.703.483.48

3.003.000.903.00

Temperature

min

303065653030303030303030303020203434404040253025

1520204040313125254060

(°C)max

3030

1701703030303030303030303040403434555555303030

110909080806060

1101107060

192 Chapter 6

Table 6.3 Continued

Material

Blueberries-

Coconut-

Grapes-

redseedless

Mulberry-

Peach-

Pineapple-

Raisins~

LegumesBroad Bean

seedsFababean

-Lentil

cotyledonshilutn

seedcoatNavy Beans

"

MeatBeef

-meatball

raw

min

3.80E-113.80E-114.60E-104.60E-104.83E-114.85E-115.80E-114.83E-112.32E-102.32E-108.00E-128.00E-125.38E-105.38E-104.00E-134.00E-13

2.06E-143.66E-073.66E-071.28E-071.28E-072.06E-141.43E-114.58E-092.06E-143.33E-083.33E-08

5.48E-12l.OOE-115.56E-102.50E-10l.OOE-11

Diffusivity(mVs)

max


1.07E-061.07E-061.07E-062.27E-072.27E-071.02E-083.17E-111.02E-084.03E-125.56E-085.56E-08

1.17E-055.56E-105.56E-103.90E-10l.OOE-11

Moisture

min

0.500.500.600.600.39

0.390.500.50

3.803.800.150.15

0.120.170.170.500.500.120.130.150.120.150.15

0.041.40

1.40

(db)max

4.004.000.600.602.35

2.353.003.00

5.005.000.600.60

1.000.370.371.001.000.240.130.150.240.400.40

2.451.40

1.40

Temperature

min

37374545305050306060303030301515

2020203030303030303535

103030

14030

(°C)max

6060

110110757070758080303050507070

6540403030505050506565

18014030

14030


Table 6.3 Continued

Material

Beef Carcassbone

fatmuscle

Broiledwaste

Bull-

Ground Beef-

heat treatedraw

Heiferheat treated

rawPepperoni

-sausageSausage

-Turkey

Model FoodsAlbumin-Flour-Bran

mixture!mixlure2mixturesAmioca

-gel

hydratedCellulose-oil-water

-Corn Starch

.

min

5.48E-125.48E-123.07E-115.83E-108.06E-068.06E-066.30E-116.30E-113.00E-113.00E-117.50E-114.00E-115.40E-111.30E-105.40E-114.70E-114.70E-114.70E-1 11.3 IE-071.3 IE-078.00E-158.00E-15

l.OOE-145.80E-101.15E-098.00E-105.80E-106.13E-116.13E-115.50E-101.40E-093.10E-093.10E-091.89E-101.89E-10

Diffusivity(mVs)

max

5.83E-105.48E-123.07E-115.83E-101.17E-051.17E-058.20E-118.20E-112.30E-103.07E-112.30E-101.70E-102.14E-102.14E-101.20E-105.70E-115.70E-115.70E-111.3 IE-071.3 IE-078.00E-158.00E-15

2.25E-081.85E-091.85E-091.26E-091.05E-097.30E-097.30E-091.20E-092.40E-093.10E-093.10E-092.60E-102.60E-10

Moisture(db)

min

0.300.300.760.760.160.160.600.601.001.001.000.19

0.190.320.320.040.04

0.020.080.080.130.100.020.020.070.05

max

0.300.300.760.761.600.161.001.601.001.001.000.19

0.190.320.320.040.04

80.000.830.820.830.800.930.930.750.75

Temperature(°C)

min

1010202025253030303030121212

1801802222

20757575752020606068683030

max

3232303075257560757560121212

1801802222

105105105105105100100606068683030

194

Table 6.3 Continued

Chapter 6

Material

Flour-

Glucoose-Starch-

Gluten-Starchgelatinized

ungelatinizedHylon-7

-gel

hydratedPolyacrylamide Gel

-Potato Starch

-Rice Starchfull heatednon-heated

Starch~

NutsAlmond

-Hazelnuts

shelledunshelled

Peanut Podshull

kernelPeanuts

.

min

3.86E-123.86E-126.60E-116.60E-111.90E-112.20E-1 11.90E-112.48E-102.48E-108.00E-101.70E-098.77E-128.77E-12l.OOE-14l.OOE-148.48E-118.48E-113.92E-101.50E-101.50E-10

4.36E-132.24E-122.24E-121.48E-091.48E-093.65E-094.36E-134.36E-138.52E-124.00E-114.00E-11

Diffusivity(m2/s)

max

3.20E-113.20E-115.90E-105.90E-104.00E-114.00E-1 13.20E-112.25E-082.25E-083.70E-092.70E-093.00E-103.00E-102.40E-112.40E-1 14.62E-092.84E-094.62E-096.90E-106.90E-10

1.24E-082.39E-122.39E-121.24E-088.39E-091.24E-082.36E-101.17E-102.36E-104.00E-114.00E-11

Moisture

min

0.070.070.600.600.090.090.090.030.030.050.050.100.100.800.800.670.670.670.600.60

0.050.050.050.150.150.150.600.600.600.100.10

(db)max

0.170.170.600.600.200.200.201.000.901.000.752.002.00

80.0080.00

4.004.002.330.600.60

0.600.050.050.150.150.150.600.600.600.100.10

Temperature

min

2525303074747420206060404025252525253030

27281281

3030302727275050

(°C)max

25255050747474

1001006060404025258080805050

281281281

8080804343435050


Table 6.3 Continued

Material

OtherCanolaembryo

kernelChocolate

darkCoffeeextract

Eggfresh

incubatedSunflower Seeds

hullkernelToriaseeds

VegetablesBeet

-Broccoli

stemsCarrot

-Cassava

rootsGarlic

-Okra

-Onion

-Paprika

-Pea

.

min

8.20E-142.10E-092.10E-093.70E-098.20E-148.20E-145.00E-115.00E-111.03E-111.03E-111.39E-117.00E-111.70E-107.00E-1 12.85E-112.85E-11

2.20E-121.50E-091.50E-091.78E-101.78E-102.20E-122.20E-121.07E-101.07E-101.14E-111.14E-114.28E-104.28E-101.38E-111.38E-115.80E-115.80E-111.10E-101.10E-10

Diffusivity(mVs)

max


3.05E-071.50E-091.50E-093.41E-093.41E-097.46E-097.46E-092.15E-092.15E-094.18E-104.18E-106.80E-096.80E-096.60E-096.60E-094.08E-104.08E-104.40E-104.40E-10

Moisture

min

0.000.000.000.002.002.00

0.801.000.800.070.070.070.080.08

0.03

2.002.000.100.100.600.600.100.102.002.000.100.10

0.500.50

(db)max

2.000.050.050.052.002.00

1.001.000.800.070.070.070.120.12

15.00

13.9313.9315.0015.000.640.641.501.502.002.00

10.0010.00

1.501.50

Temperature

min

20808080202030303333334040405050

20656525252020555522226060404025253030

(°C)max

80808080202070704040405050508080

30065659090

100100909090908080808070706565

196 Chapter 6

Table 6.3 Continued

Material

Pepper-

redpowderPigeon Pea

kernelPotato

-restructured product

tissueSoya Meal

-Soybean

-grains

Sugar Beetroots

Tapiocaroots

Tomato-

concentrate dropletsTurnip

-Yam

.

min

5.86E-111.85E-095.86E-112.88E-112.88E-118.00E-128.00E-127.30E-101.67E-091.47E-101.47E-109.30E-119.30E-119.30E-111.96E-101.96E-103.50E-103.50E-101.52E-101.52E-101.69E-107.61E-127.61E-127.30E-107.30E-10

Diffusivity(m2/s)

max

1.16E-081.16E-084.08E-107.70E-117.70E-111.25E-081.25E-084.52E-091.67E-094.01E-084.01E-081.17E-053.05E-072.15E-091.30E-091.30E-099.00E-109.00E-106.46E-092.36E-096.46E-093.62E-093.62E-091.81E-091.81E-09

Moisture(db)

min max

0.060.100.060.200.200.100.10

1.850.030.030.190.190.202.212.211.051.050.50

10.000.500.310.310.060.06

10.0010.00

0.060.200.20

10.0010.00

1.850.220.221.001.001.003.003.001.051.05

10.0010.00

0.507.007.000.140.14

TemperatureCO

min max

25602550502424

1056565652020254040555540406020204040

8080709090

18518510565

300300

95957281819797

10080

1001001005050


III. MOISTURE DIFFUSIVITY OF FOODS AS A FUNCTION OFMOISTURE CONTENT AND TEMPERATURE

Moisture diffusivity of foods depends strongly on moisture, temperature andstructure of the material. In porous materials, the porosity and the pore structuredistribution significantly affect diffusivity. Limited information concerning theeffect of structure on diffusivity is available in the literature (Chapter 5). On theother hand, the effect of moisture and temperature on diffusivity has been studiedmore extensively. Nevertheless, general models describing the effect of moisturecontent and temperature on diffusivity of foods do not exist.

A large number of empirical equations are summarized and analyzed in de-tail by Marinos-Kouris and Maroulis (1995), Zogzas et al. (1996), and Mittal(1999). According to the most popular consideration, the effect of temperature andmoisture content is introduced into the Arrhenius model.

A concept proposed by Maroulis et al. (2001) is adopted here and applied toobtain an integrated and uniform analysis of the available moisture diffusivitydata. The concept was applied simultaneously to all the data of each material, re-gardless of data sources. Thus, the results are not based on the data of only oneauthor, and consequently they are of elevated accuracy.

Assume that a material of intermediate moisture content consists of a uni-form mixture of two different materials: (a) a dried material and (b) a wet materialwith infinite moisture. Moisture diffusivity is, generally, different for each mate-rial. The diffusivity of the mixture could be estimated using a two-phase structuralmodel:

(6-1)^l + X " l + X

where D (m2/s) the effective moisture diffusivity, DXo (m2/s) the moisture diffusiv-ity of the dried material (phase a), DXj (m2/s) the moisture diffusivity of the wetmaterial (phase b), X (kg/kg db) the material moisture content, and T (°C) the ma-terial temperature.

Assume that the diffusivities of both phases depend on temperature by anArrhenius-type model:

(6-2)= u exp - —R(T T

198 Chapters

(6-3)x, , -r R , T T

where Tr= 60°C a reference temperature, R = 0.0083143 kJ/mol K, the ideal gasconstant, and D0,DI,E0,EI are adjustable parameters of the proposed model. Thereference temperature of 60°C was chosen as a typical temperature of air-drying offoods.

Thus, the moisture diffusivity for every material is characterized and de-scribed by four parameters with physical meaning:

• D0 (m2/s) diffusivity at moisture X = 0 and temperature T = Tr

• Dj (m2/s) diffusivity at moisture X = oo and temperature T - Tr

• E0 (kJ/mol) activation energy for diffusion in dry material at X = 0• Ei (kJ/mol) activation energy for diffusion in wet material at X = oo

The resulting model is summarized in Table 6.4 and can be fitted to data us-ing a nonlinear regression analysis method. The model is fitted to all literature datafor each material and the estimates of the model parameters are obtained. Then theresiduals are examined and the data with large residuals are rejected. The proce-dure is repeated until an acceptable standard deviation between experimental andcalculated values is obtained (Draper and Smith, 1981).

Among the available data only 19 materials have more than 10 data, whichcome from more than 3 publications. The procedure is applied to these data andthe results of parameter estimation are presented in Table 6.5 and in Figure 6.6. Itis clear that moisture diffusivity is larger in wet materials.


Figures 6.7-6.29 present retrieved moisture diffusivities from the literatureand model-calculated values for selected food materials as a function of moisturecontent and temperature. Moisture diffusivity D tends to increase with the mois-ture content X and the temperature T. The maxima of the curves observed at lowmoisture contents in the model foods (see Chapter 5) are smoothed out by the sta-tistical treatment of the data.

It must be noted that the regression procedure was applied simultaneously toall the data of each material, regardless of the data sources. Thus, the results arenot based on the data of only one author and consequently they are of higher accu-racy and general applicability.

The diffusivity parameters D0 and Z)/ of the proposed model, shown in Fig-ure 6.6, vary in the range of 10~10 to 10~8 m2/s. It should be noted that the self-diffusivity of water is approximately 10"9 m2/s, and the moisture diffusivity inbone-dry food material should be lower (in our analysis, by a factor of 100).

Low moisture diffusivities are found in nonporous and sugar-containingfoods, while higher values of moisture diffusivity characterize porous food materi-als. Diffusivities higher than the self-diffusivity of water are indicative of vapordiffusion in porous solids.

The moisture diffusivity increases, in general, with increasing moisture con-tent. Temperature has a positive effect, which depends strongly on the food mate-rial. The energy of activation for diffusion E of water is, in general, higher in thedry food materials. Some observed exceptions may be explained by the prevailingtype of diffusion. Thus, lower values of activation energy for diffusion are ex-pected for porous foods, where vapor diffusion is important. In general, tempera-ture has a stronger effect on diffusivity in the liquids and solids than in the gasstate (see Chapter 2).

200 Chapter 6

Table 6.4 Mathematical Model for Calculating Moisture Diffusivity in Foods as aFunction of Moisture Content and Temperature

Proposed Mathematical Model

£> = •l + X

Do exp

where

R T TX

\+x t expRT T

D (m2/s) the moisture diffusivity,X (kg/kg db) the material moisture content,r(°C) the material temperature,Tr = 60°C a reference temperature, andR = 0.0083143 kJ/mol K the ideal gas constant.

Adjustable Model Parameters

• D0 (m2/s) diffusivity at moisture X = 0 and temperature T = Tr

• D, (m2/s) diffusivity at moisture X = oo and temperature T = T.• E0 (kJ/mol) activation energy for diffusion in dry material at X = 0• EI (kJ/mol) activation energy for diffusion in wet material at X = oo


Table 6.5 Parameter Estimates of the Proposed Mathematical Model

Material No. of No. ofPapers Data

Cereal productsCorn

-dent

grainskernel

pericarpPasta

-Rice

kernelRough rice

-Wheat

-Fruits

Apple-

Banana-

GrapesseedlessRaisins

-Model foods

Amioca-

Hvlon-7-

VegetablesCarrot

-Garlic

-Onion

-Potato

-

43343

3

3

7

6

8

4

3

3

4

5

9

4

4

16

2615282513

21

12

35

22

39

34

32

10

49

48

90

22

31

106

Di(m2/s)

4.40E-091.19E-081.15E-095.87E-101.13E-09

1.39E-09

9.75E-09

2.27E-09

1.94E-09

7.97E-10

2.03E-09

5.35E-09

8.11E-10

1.52E-08

1.96E-08

2.47E-09

5.33E-10

1.45E-08

1.57E-09

Do Ei Eo(mVs) (kJ/mol) (kJ/mol)

O.OOE+00O.OOE+006.66E-115.32E-10

O.OOE+00

O.OOE+00

O.OOE+00

O.OOE+00

1.30E-09

1.16E-10

4.66E-10

O.OOE+00

1.05E-10

1.52E-08

1.96E-08

1.54E-09

1.68E-11

O.OOE+00

4.31E-10

0.049.410.20.0

10.0

16.2

12.5

12.7

0.0

16.7

9.9

34.0

21.4

0.0

0.0

13.9

15.4

70.2

44.7

10.473.157.833,8

5.0

2.0

2.0

0.7

46.3

56.6

4.6

10.4

50.1

33.3

24.2

11.3

7.1

10.4

76.9

sd(m2/s)

1.48E-103.30E-103.17E-101.88E-112.34E-11

7.71E-12

5.52E-11

3.66E-11

9.53E-11

1.92E-10

1.77E-10

1.45E-10

6.88E-11

1.02E-09

3.87E-09

1.69E-09

7.43E-11

1.58E-09

4.02E-10

202 Chapter 6

l.E-06

l.E-07

S" l.E-08S,

•I" l.E-09

IS l.E-10

l.E-11

l.E-12

• Moisture - infiniteQ Moisture - zero

8 1 1 1£ o- JJ3 | «

T S .a a a= t •= •! ai a o o ,2

100

oEi-:

>.£*

I.

• Moisture = infinite13 Moisture = zero

* I 1o o S.

Figure 6.6 Parameter estimates of the proposed mathematical model.


l.E-06

l.E-07

Moisture (kg/kg db)

Figure 6.7 Predicted values of moisture diffusivity of model foods at 25°C.

204 Chapter 6

.E-06

.E-07

l.E-08

.|" l.E-09

l.E-10

.E-l l

l.E-12

Hylon-7

— Amioca —

H Model foods | ——

:ratui

i 1

•e (°C) = 60 -f—

0.1 1Moisture (kg/kg db)

10

Figure 6.8 Predicted values of moisture diffusivity of model foods at 60°C.


l.E-06

l.E-07

l.E-08

f l.E-09 4

l.E-10

l .E-1

l.E-12

Moisture (kg/kg db)

Figure 6.9 Predicted values of moisture diffusivity of fruits at 25°C.

206 Chapter 6

l.E-06

l.E-07

Temperature (°C) = 60 -4

Moisture (kg/kg db)

Figure 6.10 Predicted values of moisture diffusivity of fruits at 60°C.


l.E-06

l.E-07

1 HTempera

Vegetables

ture i °C) = 25

l.E-08

l.E-120.1 10

Moisture (kg/kg db)

Figure 6.11 Predicted values of moisture diffusivity of vegetables at 25°C.

208 Chapter 6

l.E-06

l.E-07

l.E-120.1 1

Moisture (kg/kg db)

10

Figure 6.12 Predicted values of moisture diffusivity of vegetables at 60°C.


l.E-06

l.E-07

l.E-120.1 10

Moisture (kg/kg db)

Figure 6.13 Predicted values of moisture diffusivity of corn at 25°C.

210 Chapter 6

l.E-06

.E-07

-h- r

—— |Cereal products (corn)

Temperature (°C) = 60

Jtrf

l.E-120.1

Moisture (kg/kg db)

Figure 6.14 Predicted values of moisture diffusivity of corn at 60°C.


Garlfc.E-06

.E-07 -

.E-08 -

.E-09 -

.E-10 -j

——

——————— | C(

— Rice kernel

^^r^j i•" ^ j

•"

real products

Temperature (°C) = 25

i

t~r

i

-

Corn

Vheat

Lx ' Rough rice ! '

.E - l l -

F.n a

Xx^|x-F—— ^^r ——————

^^^ Pact n

asta

i

0.1 1Moisture (kg/kg db)

10

Figure 6.15 Predicted values of moisture diffusivity of cereal products at 25°C.

212 Chapter 6

l.E-06

.E-07

l.E-08

£

£ l.E-09

1 Cereal products^«™-™-™M™«»™I»™Temperature (°C) = 60

' Rice kernel

l.E-10 -I wheat

l .E-11

Rough rice

Pasta

l.E-120.1 1

Moisture (kg/kg db)

10

Figure 6.16 Predicted values of moisture diffusivity of cereal products at 60°C.


Fruits AppleTotal Number of Papers 8

Total Experimental Points 64Points Used in Regression Analysis 36

Standard Deviation (sd, rti'/s) 1.92E-10Relative Standard Deviation (rsd, %) 457

(56%)

Parameter EstimatesDi (m2/s)Do (mj/s)

Ei (kJ/mol)Eo (kJ/mol)

7.97E-101.16E-10

16.756.6

.E-06

l.E-07

l.E-08

•I" l.E-09

l.E-10

l.E-11

Temperature (°C)— 140— «60

A 80

l.E-120.1 1.0

Moisture (kg/kg db)10.0

Figure 6.17 Moisture diffusivity of apple at various temperatures and moisturecontents.

214 Chapter 6

Fruits GrapesTotal Number of Papers 3


Standard Deviation (sd, rrrVs) 1.45E-10Relative Standard Deviation (rsd, %) 1 3 1

seedless

(63%)

Parameter EstimatesDi(m2/s) 5.35E-10

Do (m2/s) O.OOE+00Ei (kJ/mol) 34.0Eo(kJAnol) 10.4

1 F-Ofi ————————————————— ; ——————

l.E-07 -

l.E-08 -

•f l.E-09 -11

l.E-10 -

l.E-11 -

l.E-12 ————————————————————————————————————

—————————————————————

1

——————————

h-—————————— ——————

i

1

^

^^=1J^

>•

!•1

IEr^

^

••

i

— Tem

-1

^

•**

01

I'-

perat• 40• 60A 80

ure (°C)—

=^~~~~

-_

~* — » ———

—_L

^

0.1 1.0 10.0Moisture (kg/kg db)

Figure 6.18 Moisture diffusivity of grapes at various temperatures and moisturecontents.


Fruits BananaTotal Number of Papers 4

Total Experimental Points 49Points Used in Regression Analysis 15 (31%)

Standard Deviation (sd, rrvVs) 1.77E-10Relative Standard Deviation (rsd, %) 1 5

Parameter EstimatesDi (m'Vs) 2.03E-09Do(m'Vs) 4.66E-10

Ei (kJ/mol) 9.9Eo (kJ/mol) 4.6

l.F-06 ———————————

l.E-07 -

l.E-08 -

Diff

usiv

ity (m

2 /s)

hn

m^-

o

0

MD

l.E-11 -

l.E-12 -0

h=Hs_^M

'

^s.BH

—— Tern perature (°C) —

• 60-4- A 80 ———

Si^

•

(

——— ———

iSm.

-1 ————I

—— f ———"

*—— ———

| —

1 1.0 10.0Moisture (kg/kg db)

Figure 6.19 Moisture diffusivity of banana at various temperatures and moisturecontents.

216 Chapter 6

Vegetables PotatoTotal Number of Papers 13


Standard Deviation (sd, m2/s) 4.02E-10Relative Standard Deviation (rsd, %)______122

(45%)

Parameter EstimatesDi (m2/s)Do (m2/s)


1.57E-094.31E-10

44.776.9

l.E-06

l.E-07

Temperature ( C)• 40• 60A SO

l.E-120.1 1.0


Figure 6.20 Moisture difrusivity of potato at various temperatures and moisturecontents.


Vegetables CarrotTotal Number of Papers 12


Standard Deviation (sd, m2/s) 1.69E-09Relative Standard Deviation (rsd, %)_____18699

(92%)

Parameter EstimatesDi (m"/s)Do(rrrVs)

Ei (kJ/mol)Eo(kJ/mol)

2.47E-091.54E-09

13.911.3

I .F-Ofi

l.E-07

l.E-08

• l.E-09

l.E-10

l.E-11

l.E-120.1 1.0


Figure 6.21 Moisture diffusivity of carrot at various temperatures and moisturecontents.

218 Chapter 6

Vegetables OnionTotal Number of Papers 4


Standard Deviation (sd, m'Vs) 1.58E-09Relative Standard Deviation (rsd, %) 575

(71%)

Parameter EstimatesDi (m"/s)Do (mVs)


1.45E-09O.OOE+00

70.210.4

1.E-06

l.E-0.1 1.0


Figure 6.22 Moisture diffusivity of onion at various temperatures and moisturecontents.


Vegetables GarlicTotal Number of Papers 4


Standard Deviation (sd, m'Vs) 7.43E-1 1Relative Standard Deviation (rsd, %) 385

Parameter EstimatesDi (m"/s) 5.33E-10Do(nWs) 1.68E-11

Ei(kJ/moI) 15.4Eo(kJ/mol) 7.1

l.E-06 -

l.E-07 -

l.E-08 -

W5

"E,•f l.E-09 -

la5

l.E-10 ,1

l.E-11 -

l.E-12 - —— — — - - - - - -

*~\fff*^

*•?*—••if*

^*^ ——

———— • —————

^•*

«•~-}«•3-

— Tern

11

r-r*~mi~~*

perat• 40• 60A 80

ure(°C) =— j —re=P


Figure 6.23 Moisture diffusivity of garlic at various temperatures and moisturecontents.

220 Chapter 6

Cereal Products WheatTotal Number of Papers 5


Standard Deviation (sd, m2/s) 9.53E-1 1Relative Standard Deviation (rsd, %) 54

Parameter EstimatesDi(m"/s) 1.94E-10

Do(mVs) 1.30E-10Ei (kJ/mol) 0.0Eo (kJ/mol) 46.3

1 ,F,-06 i ————— i —————

l.E-07 -

l.E-08 -

tfi

\

•1" l.E-09 -V)

S

l.E-10 -

l.E-11 -

l.E-12 -

!

i

1

——— 4,——— *-•

———— B40 ———• 60A 80 ———

j

k A I

^J

———— I

— •-1

1

EE^

\

\»

fc — f—1

111 ——

i

)

i -


Figure 6.24 Moisture diffusivity of wheat at various temperatures and moisturecontents.


Cereal Products Corn dentTotal Number of Papers 3


Standard Deviation (sd, rrrVs) 3.30E-10Relative Standard Deviation (rsd, %) 343

Parameter EstimatesDi(rrrVs) 1.19E-09Do (m"/s) O.OOE+00

Ei (kJ/mol) 49.4Eo(kJ/mol) 73.1

l.E-06 -

l.E-07 -

l.E-08 -

5"=

•f l.E-09 -M

a

l.E-10 Ji11

l.E-11 -

i

- _>1- . T —*+*

**?—- ———— < k —\±*L****T

•^M; j^^\^^

\^^

^*

— «1

i

t-1

peral• 40• 60A80

ure CC)

0.1 1.0 10.0

Moisture (kg/kg db)

Figure 6.25 Moisture diffusivity of corn (dent) at various temperatures and mois-ture contents.

222 Chapter 6

Cereal Products Corn grainsTotal Number of Papers 3


Standard Deviation (sd, m'Vs) 3 . 1 7E- 1 0Relative Standard Deviation (rsd, %) 1 53

Parameter EstimatesDi(m<Vs) 1.15E-09

Do (m'Vs) 6.66E-11Ei (kJ/mol) 10.2Eo(kJ/mol) 57.8

l.F-06 —————————————————————————————————— , —

l.E-07 -

l.E-08 -

"E•f l.E-09 -

1

1l.E-10 -

l.E-11 T

•

1

r^

k —

• —

li

TAmnorn+iifo ("f~

Jj

)

———— «40 ——— - -• 60

————— A 80 —p-

• "* | =>

I

\

1

u-i

3


Figure 6.26 Moisture diffiisivity of corn (grains) at various temperatures andmoisture contents.


Cereal Products Corn kernelTotal Number of Papers 3


Standard Deviation (sd, nWs) 1.88E-1 1Relative Standard Deviation (rsd, %) 32

Parameter EstimatesDi(m'Vs) 5.87E-11Do(m'Vs) 5.32E-11


I P-flfi ————————————————— r-i ————————————————————————

l.E-07 -

l.E-08 -

•f l.E-09 -ia

l.E-10 -

l.E-11 -

l.E-12 -

-f-4

=^1• J^^

———— B40 —• 60

———— A 80 —

CO =EtE~

It4=^_•

0.1 1.0Moisture (kg/kg db)

q

10.0

Figure 6.27 Moisture diffusivity of corn (kernel) at various temperatures andmoisture contents.

224 Chapter 6

Cereal Products Corn pericarpTotal Number of Papers 3


Standard Deviation (sd, nTVs) 2.34E-1 1Relative Standard Deviation (rsd, %) 7558

Parameter EstimatesDi(m2/s) 1.13E-10Do (nrVs) O.OOE+00


l.E-06 -

l.E-07 -

l.E-08 -

ff>(S

•f l.E-09 -•aQ

l.E-10 -

l.E-11 '

l.E-12 -0

r =

—————i •—— i| ^*

1

:

—— Temperati

ure(°C) =g=- ———— «40

• 60-p ——— A 80 —

i

^

q

i

'


Figure 6.28 Moisture diffusivity of corn (pericarp) at various temperatures andmoisture contents.


Cereal Products PastaTotal Number of Papers 3


Standard Deviation (sd, m2/s) 7.71E-12Relative Standard Deviation (rsd, %) 36

Parameter EstimatesDi(m2/s) 1.39E-10Do(m2/s) -1.6 IE-21

Ei(kJ/mol) 16.2Eo (kJ/mol) 2.0

i F-Ofi ———————

l.E-07 -

l.E-08 -

Sflts

•f l.E-09 -'&

3

l.E-10 -

-jfr-^**^&k=^TJ

l .E-l l ^

'

l.E-12 -0

• ————

*—

—— Tern perat• 40• 60A 80

~ 1 _ f

ure(°C) =t4=_ . ^rj^


Figure 6.29 Moisture diffusivity of pasta at various temperatures and moisturecontents.

226 Chapter 6

REFERENCES

Abdelhaq, E.H., Labuza, T.P., 1987. Air Drying Characteristics of Apricots. Jour-nal of Food Science 52:342-345.

Afzal, T.M., Abe, T., 1998. Diffusion in Potato During Far Infrared RadiationDrying. Journal of Food Engineering 37:353-365.

Alvarez, P., Shene, C., 1996. Experimental Study of the Heat and Mass Transferduring Drying in a Fluidized Bed Dryer. Drying Technology 14:701-718.

Alvarez, P.I., Blasco, R., 1999. Pneumatic Drying of Meals: Application of theVariable Diffusivity Model. Drying Technology 17:791-808.

Alvarez, P.I., Legues, P., 1986. A semi-Theoritical Model for the Drying ofThompson Seedless Grapes. Drying Technology 4:1-17.

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Takeuchi, S., Maeda, M., Gomi, Y., Fukuoka, M., Watanabe, H., 1997. TheChange of Moisture Distribution in a Rice Grain during Boiling as Observedby NMR Imaging. Journal of Food Engineering 33:281-297.

Tang, J., Sokhansanj, S., 1993. Moisture Diffusivity in Laird Lentil Seed Compo-nents. Trans. of the ASAE 36:1791-1798.

Teixeira, M.B.F, Tobinaga, S., 1995. Theoretical and Experimental Study of Wa-ter Transport in a Hollow Cylinder Applied to the Drying of Round SquidMantle. Drying Technology 13:2069-2081.

Teixeira, M.B.F., Tobinaga, S., 1998. A Diffusion Model for Describing WaterTransport in Round Squid Mantle During Drying with a Moisture-dependentEffective Diffusivity. Journal of Food Engineering 36:169-181.

Thakor, N.J., Sokhansanj, S., Sosulski, F.W., Yannacopoulos, S., 1999. Mass andDimensional Changes of Single Canola Kernels during Drying. Journal ofFood Engineering 40:153-160.

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236 Chapter 6

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Diffusivity and Permeability of SmallSolutes in Food Systems

I. INTRODUCTION

The diffusion of small molecules in food and food packaging materials isimportant in food processing operations, food product development, and foodprocess control. It involves the transport of small molecules, such as sugars, or-ganic acids, flavor components, salts, preservatives, and gases (e.g. oxygen, car-bon dioxide). In addition, the transport of larger molecules, such as lipids, is ofinterest to food processing, such as oil extraction. The transport of solutes is offundamental importance to the physical separation processes, such as solvent ex-traction (e.g. sugar/water), ion exchange (e.g. de-acidification and de-bittering ofjuices), reverse osmosis, and ultrafiltration. Mass diffusivity is the basic compo-nent of permeability of packaging films and food coatings, which are used as bar-riers to water, oxygen, and carbon dioxide transport in food materials.

Theoretical prediction of solute diffusivity in solid and semisolid food mate-rials is not feasible, and experimental measurements and data are necessary. Theexperimental methods and theoretical analysis of water transport in food materials(Chapter 5) are utilized in the diffusion of small solutes in food systems.

A. Diffusivity of Small SolutesMass transfer of small molecules can be analyzed by either the diffusion

(Pick) model or the mass transfer coefficient concept (Cussler, 1997). The diffu-sion model, in general, is used in modeling mass transport of water and small sol-utes within foods, since solid and semisolid food materials have complex struc-tures that strongly effect mass diffusivity. Mass transfer coefficients are used in

237

238 Chapter 7

designing processing equipment where mass is transported between phases, as indrying and separation processes, such as extraction and membrane processing.

The transport of small solutes in simple gases and liquids can be predictedby molecular dynamics (see Chapter 2), and reliable data on mass diffusivity isavailable in the literature. Prediction of diffusivity is difficult in complex fluidfoods and in solid/semisolid foods, where experimental measurements and empiri-cal correlations are essential.

Although small solutes can be transported in food systems by differentmechanisms, molecular diffusion is generally accepted as the basic transport proc-ess, in a similar manner with the transport of water (see Chapter 5). Thus, the ef-fective diffusivity, an overall transport coefficient, can be defined, assuming thatthe driving force is a concentration gradient (dC/dz), and applying the diffusion(Pick) equation:

(8C/&) = (d/dz) [D(cC/&)} (7-1)

Table 7.1 shows some typical values of diffusivity in gases, liquids, and sol-ids (see also Table 2-4).

The diffusivity of solutes in fluid foods is of the order of 1x10"9 m2/s, whilein solid and semisolid foods the diffusivity varies widely in the range 10'14 to 10~8

m2/s, due to heterogeneous structure, which involves diffusion in gas and liquidphases. Diffusion in polymers varies from 10"18 to 10'10 m2/s, due to differentstructures (rubbery and glassy states).

Table 7.1. Typical Diffusivities of Small Solutes

Diffusivity in D, m / sGases IxlO'5

Liquids IxlO'9

Polymers (rubbery) IxlO'14 - IxlO'10

Polymers (glassy) IxlO'18 - IxlO'12

Diffusivity and Permeability of Small Solutes in Food Systems 239

B. Measurement of DiffusivityThe measurement of diffusivities of small molecules in solid food materials

is discussed in Chapter 5, in connection with the transport of water (moisture).Some of these methods are also used for the determination of the diffusivities ofsmall solutes in polymers and food solids, especially the sorption kinetics, thepermeability and the concentration-distribution techniques. Additionally, twomore experimental methods, used for the measurement of diffusivities of solutes ingases, liquids, and membranes, are important for mass transfer in food systems,i.e. the diaphragm cell and the Taylor dispersion method.

1. The Diaphragm CellThe Stokes diaphragm cell is used for the measurement of the diffusion co-

efficients in gases, liquids, and across membranes, with an accuracy of up to 0.2%(Cussler, 1997). The method is based on the diffusion of the solute between twocompartments separated by either a fritted glass surface or by a porous membrane,as shown diagrammatically in Figure 7.1.

The two compartments are kept at different but constant solute concentra-tions, using magnetic stirring. The fritted glass diaphragm may be replaced by apiece of filter paper for faster measurement.

AB

magneticstirrer

glass"frit

stirringbars

Figure 7.1 Stokes diaphragm cell.

240 Chapter 7

After a certain time t of diffusion, the contents of the two compartments areanalyzed for the concentration of the solute, and the diffusivity D is estimatedfrom the equation:

D = (I/ft) [(C, o - C2 o)/(Cj - C2)] (7-2)

where C/o, C?ft C/ C2 are, respectively, the initial and final solute concentrations incompartments 1 and 2, and jBis the calibration constant, related to the dimensionsof the cell:

P=(Alt)(\IV,+ \/V2) (7-3)

where A is the area available for diffusion, / is effective thickness of the dia-phragm, and F; and V2 are the volumes of the two cell compartments.

The two compartments are placed vertically, so that the diaphragm surfacefor diffusion is horizontal. Usually, 1 and 2 indicate the bottom and top cell com-partments, respectively. The horizontal position of the diaphragm is necessary toassure uniform concentration gradient and prevent free convection, which mightdevelop in a vertical or inclined position. The accuracy of the method depends onthe accuracy of determination of the concentration differences between the com-partments, and not on the concentrations themselves.

2. The Taylor Dispersion MethodThe Taylor dispersion method, used for both gases and liquids, is based on

the dispersion by diffusion of a solute injected in a stream of a carrying fluid (Cus-sler, 1997). A sharp pulse on the solute is injected into a fluid moving in laminarflow in a long tube. The flow exiting the tube is analyzed for the solute (by differ-ential refractometry) over a period of time and the concentration profile is deter-mined. The diffusivity D is estimated from the concentration profile C at time t,representing the decay of a pulse, using the equation:

C = (Ml nr ') exp[-(z - u t)2 / 4 E t] I (4 nEi )1/2 (7-4)

where Mis the total solute injected, r is the tube radius, z is the diffusion distance,and £ is a dispersion coefficient given by the equation:

E = (u r)21 (48 D) (7-5)

where u is the average velocity of the flowing solvent. Since the refractive index isa linear function of concentration, the refractive index profile can be used for thedetermination of the diffusivity.


Very high accuracies in the determination of diffusivity of solutes in fluidscan be obtained by interferometers, such as the Gouy, the Mach-Zehnder, and theRayleigh instruments. The interferometers are based on measuring an unsteady-state profile of the refractive index of two solutions in a transparent system, diffus-ing into each other.

II. DIFFUSIVITY IN FLUID FOODS

The diffusivity of solutes in dilute aqueous solutions is of importance tofood systems, since most food components are present in foods at low concentra-tions (infinitely dilute solutions).

A. Dilute SolutionsTable 7.2 shows some typical diffusivities of solute gases in dilute aqueous

solutions (Cussler, 1997) (see also Table 2-4). The diffusivity of water and oxygenin dilute ethanol solutions at 25°C is 1.24 x 1Q"9 and 2.64 x 10~9m2/s, respectively.

Table 7.3 shows some typical diffusivities of solutes in dilute water solu-tions, which are of interest to food systems (Cussler, 1997; Schwartzberg andChao, 1982).

The diffusivity of low-molecular weight solutes is in the same range withthe self-diffusivity of water (1 x 1Q"9 m2/s). The diffusivity, in general, decreasesas the molecular size of the solute is increased. High-molecular weight food com-ponents, such as proteins and polysaccharides, have diffusivities close to that ofwater in a solid starch/sugar gel (see Chapter 5).

Table 7.2 Diffusivities of Gases in Dilute WaterSolutions at 25°CSolute________________Ax 10'9m2/sAir 2.00Oxygen 2.10Nitrogen 1.90Chlorine 1.25Carbon dioxide 1.90Ethylene 1.87Hydrogen 4.50Methane 1.49Ammonia 1.64

242 Chapter 7

Table 7.3 Diffusivities of Solutes in DiluteWater Solutions at 25°C

SoluteEthanolAcetic acidButyric acidGlycineSucroseGlucose/fructoseMaltoseGlycerolHemoglobinFibrinogenLactoglobulinOvalbumin

D, x 10-'°m2/s8.4012.19.2010.65.406.904.809.200.690.200.700.78

The diffusivity of solutes in dilute water solutions can be predicted by em-pirical equations based on molecular dynamics and hydrodynamics, like theWilke-Chang equation (2-34) and the Stokes-Einstein equation (7-6):

D = (kBr>/(6xtiBr) (7-6)

where r is the particle radius, rjB is the viscosity of the solvent (water), T is theabsolute temperature, and kB = 1.38xlO"22 J/molecule K is the Boltzmann constant.

The Stokes-Einstein equation is based on hydrodynamic and not molecularforces, and it is applicable to solutes of molecular size five times larger than thesolvent. For smaller molecules, the Wilke-Chang equation gives better prediction(Cussler, 1997). In both equations, the diffusivity is inversely proportional to theviscosity of the solution. In very viscous solutions, the diffusivity becomes inde-pendent of viscosity, e.g. the D of sugar in a gel is nearly equal to the D in water.

B. Concentrated SolutionsThe diffusivity of solutes in liquids D varies considerably with the concen-

tration, sometimes with maximum or minimum values at certain concentrations.The D can be estimated from the diffusivity at infinite dilution Dm using a correc-tion factor to account for the effect of chemical activity on the transport rate (Reidetal., 1987; Cussler, 1997):

D = D0(l+dlna/dlnC) (7-7)


where a is the activity and C is the concentration of the solute in the solution.The diffiisivity of the mixture at infinite dilution D0 can be estimated from

the diffusivities at infinite dilution of the solute and the solvent, and the corre-sponding mole fractions (x/ and *?):

A, = [A,(x,= i)]MZ)0fe=im (7-8

The correction factor (d lnor/9 InQ represents the molecular and hydrodynamicinteractions in the concentrated solution, and it is negative in nonideal solutions(Cussler, 1997). Thus, D of the solute in a mixture becomes lower than D0 at bothextreme concentrations (xlt x 2 = 1), with a minimum at an intermediate concentra-tion.

III. DIFFUSION IN POLYMERS

The sorption and transport of small molecules (solutes) in polymeric materi-als are the basic physical phenomena of several important applications, such asseparation processes, barrier films, and controlled release. Most of the researchand theory in this area concerns synthetic polymers of known composition andstructure, but the available knowledge can be applied to natural polymers, whichare the basic structural components of most food materials.

Molecular (Fickian) diffusion is assumed as the main mass transport mecha-nism, although in some cases other mechanisms may be involved. Solution of thediffusion equation (7-1) forms the basis of mathematical analysis of the experi-mental data. Most of the diffusivity data of solutes in polymers have been obtainedusing the sorption and/or the permeability methods (Chapter 5).

The physical and transport properties of polymers are affected strongly bythe size and shape (linear, branched, cross-linked) of the molecules (van Krevelen,1990; Bicerano, 1996). Polymer materials can change their size (molecularweight) and microstructure during processing, changing their thermodynamic andtransport properties, such as phase equilibria and diffusion coefficients. Thesechanges should be considered in modeling and simulations of industrial processingand applications of polymers (Bokis et al., 1999).

The polymer structure is defined by the chemical constitution, set by synthe-sis (or biosynthesis) and the morphology (microstructure), set by processing(Theodorou, 1996). Quantitative relations can be established between polymerstructure and transport properties (diffusivity, permeability), based mainly on ex-perimental measurements and phenomenological correlations from various sys-tems (Petropoulos, 1994). Theoretical predictions and computer simulations, basedon molecular science, are still at the development stage, and they could find usefulapplications in the future.

244 Chapter 7

A. Diffusivity of Small Solutes in PolymersThe transport of small solutes (penetrants) normally obeys the Pick diffusion

equation, and an effective diffusivity D can be estimated, assuming that the driv-ing force is the concentration gradient. The Fickian diffusion is applicable to lowconcentrations (infinite dilution) of the solute, which is the case of most applica-tions in polymer and food systems.

In some biological systems, the thermodynamic diffusivity DT is used, basedon the chemical potential gradient, which is related to the normal diffusivity D bythe equation.

D = DT(d\na/d\nQ (7-9)

where a is the chemical activity of the species at concentration C (Frisch andStern, 1983).

In most food-related applications, the concentration of the solutes in thepolymer matrix is low, and the two coefficients become equal (D = D-f). Sorptionkinetics and permeability measurements (see Chapter 5) can be used for the de-termination of diffusivity D of solutes in polymeric materials (Vieth, 1991)

Solid polymers are amorphous materials, which exist in two nonequilibriumstates, i.e. glassy and rubbery, with transition between the states at the glass transi-tion temperature (Tg). The glassy state is characterized by a dense, tough, and lowporosity (2-8%) structure. The diffusivity of small solutes in the glassy state isvery low, e.g. IxlO"18 to IxlO"10 m2/s, depending on the polymer structure and themolecular size and concentration of the penetrant. The solute diffusivity increasessubstantially at higher solute concentrations, by plasticization of the polymer ma-trix. The activation energy for diffusion is much higher than in the rubbery state,and it increases near the glass transition temperature.

In the rubbery state, polymers are flexible, elastic materials, with relativelylarge free volume, which facilitates molecular diffusion. Crystallization or stretch-ing (induced orientation) of the polymers can reduce solute diffusivity. Liquidsand vapors may cause swelling of the glass polymeric matrix, facilitating the dif-fusion process.

Normal diffusion in the glassy and rubbery state is Fickian, i.e. the diffusionrate is proportional to the square root of time, according to Eq. (5-4) (Peppas andBrannon-Peppas, 1994). In glassy polymers (T < Tg), non-Fickian or anomalousdiffusion of solutes may take place, since diffusion and polymer relaxation arecomparable. Case II diffusion is also possible, when the diffusion rate is muchfaster than the relaxation of polymer molecules.


l.E-09 T

l.E-1430 35

Temperature (°C)

40

Figure 7.2 Arrhenius plots of diffusivity of solutes (water and carbon dioxide) in a poly-meric material showing breaks at the glass transition temperature (Tg).

Most of the research and development in polymer science and engineering isdirected to the design of specific polymer structures of known barrier properties(membranes), which can be used in separation processes of various molecularspecies. Separations based on molecular or particle size include reverse osmosis,gas separation, and ultrafiltration.

246 Chapter 7

B. Glass TransitionMass transport (diffusion) of solutes in polymers is affected strongly by the

thermodynamic state of the material. The molten polymer is a viscous fluid ofnon-Newtonian characteristics, which upon cooling forms two amorphous solidstates, the rubbery, and, at lower temperature, the glassy state. The glass transitiontemperature Tg, a second-order transformation, is an important characteristic of thepolymeric materials (Roos, 1992).

The nonequilibrium rubbery and glassy states are affected strongly by thepresence of solutes, such as gases, water and organic solvents, which reduce, ingeneral, the glass transition temperature.

The mechanical and transport properties of polymers at temperatures belowand much above Tg, are affected by the temperature, following the familiar Ar-rhenius equation. However, in the temperature range Tg to (rg+100°C) the Wil-liams-Landel-Ferry (WLF) equation is more appropriate (Levine and Slade, 1992):

log (ar) = [-C, (T - T,)} I [ C , + (T- Tg)] (7-10)

where aT is a scaling parameter, or the property ratio at T and Tgi e.g. relaxationtime, viscosity, or diffusivity, and C/ and C2 are characteristic parameters of theWLF equation, determined experimentally. In normal systems the values C\ =17.44 and C2 = 51.6 are used.

The WLF equation predicts a sharp change of the scaling factor as the tem-perature is increased immediately above Tg, e.g. the viscosity decreases 3 to 5 or-ders of magnitude at temperatures 20-3 0°C above Tg. The WLF equation can beused to nonpolymer systems, which exhibit a glass transition temperature, such assugar solutions, which are of interest to foods (Roos, 1992)

The effect of water on the glass transition temperature of polymers and otherfood components, exhibiting glass transition, is of particular importance to foodprocessing and food quality. The Tg of dry food components is relatively high butit decreases continuously even below 0°C as the moisture content is increased.Figure 7.3 schematically shows the change of Tg of a food biopolymer as a func-tion of moisture content (Roos, 1992).

Diffusivity and Permeability of Small Solutes in Food Systems 247cm

p cra

turc

(°C

)

-n

o!

!3

0

C

0 - r -

V>w\.

\

\XX.x^

—————-

x^^\^^

%v^ ——i0 5 10 15 2

Moisture (%)

Figure 7.3 Change of glass transition temperature Tg of maltodextrin with water content.

C. Clustering of Solutes in PolymersClustering of solute molecules in polymeric materials is of importance to the

sorption and diffusion properties of the system. The clustering of water is of par-ticular interest to food systems. The clustering theory of Zimm and Lundberg isbased on the statistical mechanics of fluctuations, and a simplified version of clus-tering of water in polymers is presented by Vieth (1991). The theory interprets thesorption isotherm over the entire range of penetrant activities.

The clustering function CF is a characteristic quantity that enables the calcu-lation of the tendency of the (water) molecules to cluster in the given polymermatrix. The clustering function is defined as the ratio CF = Gu/V,, where G// isthe cluster integral, calculated from the molecular pair distribution, and V\ is thepartial molecular volume of the solute (e.g. water).

The cluster function varies normally from -1 to above 2. Positive CF meansthat the solute increases the free volume of the polymer matrix, increasing thesorption capacity, diffusivity, and permeability (high relative humidity RH). Nega-tive CF means that the solute molecules are attached to specific sites dispersedthroughout the polymer matrix, reducing the sorption and transport properties (lowRH). Clustering of water can occur even at low RH by cross-linking of the poly-mer, or by the addition of plasticizers, like polyols.

248 Chapter 7

D. Prediction of DiffusivityThe experimental data of diffusivity of small solutes in polymers are often

correlated by empirical equations as a function of concentration and temperature,in a similar manner with the data on moisture diffusivity (see Chapters 5 and 6).Although satisfactory prediction is presently not feasible, some theoretical ap-proaches have been used for this purpose, i.e. the dual-sorption model, the free-volume model, and the molecular simulation method.

1. Dual-Sorption ModelThis model has been applied to the sorption and diffusion of small mole-

cules (mainly gases) in glassy polymers. The glassy matrix is assumed to containsome microcavities or "holes", created when the polymer melt or rubber isquenched (cooled rapidly). The solute is dissolved in the glassy polymer by twoparallel mechanisms, i.e. dissolution in the polymer mass according to the Henrylaw, and filling of the "holes" according to the Langmuir model (Frisch and Stern,1983; Vieth, 1991).

The Henry law for dissolution is written in the form:

CD = SDp (7-11)

where CD is the concentration of the solute in the polymer, p is the partial pressureof the solute (gas), and SD is the solubility, which is equal to \/H, where H is theHenry constant.

The Lagmuir equation for filling the holes takes the form:

CH = (C'bp)l(\+bp) (7-12)

where C'is a "hole saturation" constant, and b is a "hole affinity constant", repre-senting the ratio of rate constants of gas adsorption and desorption in microcavi-ties.

The two populations are assumed to be in local equilibrium, and the overallsolubility Sp, derived from the last equation, is given by:

S p = C / P = SD + ( C ' b ) / ( l + bp) (7-13)

The effective diffusivity D and the solubility S of the solute in the polymerare determined experimentally from sorption and permeability measurements (seeChapter 5). The effective diffusivity D is related to the diffusivities in the dis-solved state DD and in the holes DH by the overall flux equation:

J= - D (dCI dz) = - DD(dCDl dz) - DH (dCHl dz) (7-14)


The dissolved solute can diffuse readily, while only part of the solute in the"holes" is available for diffusion, i.e. DD > DH (partial-immobilization model).

2. Free- Volume ModelFree-volume models have been proposed for the prediction of transport

properties in liquids and solids, based on the availability of elements of free vol-ume within the material, through which the solute molecules can be transported(Frisch and Stern, 1983: Petropoulos, 1994). For polymeric materials, the Vrentasand Duda model, which can be used for both the glassy and the rubbery state, isdiscussed briefly here (Duda and Zielinski, 1 996).

The self-diffusion coefficient of a molecule (1) in a binary mixture is an ex-ponential function of the ratio of the volume required for diffusion of one mole V \to the total free ("hole") volume per diffusing mole VFH. The diffusion coefficientDI of a solute (1) in a binary polymer (2) mixture, in the rubbery state, is given bythe equation:

D, = Do exp(- E I RT) exp { - [ y(a>, V* , + w^ V\}\ I VFH } (7- 1 5)

where D0 is a constant, E is the activation energy, R is the gas constant, T is theabsolute temperature, coj and ca2 are the mass fractions of 1 and 2, f=F*; MjV^M^and MI and A/? are the molecular weights of 1 and 2. The accommodation factory is taken between 0.5 and 1 .0.

The specific free- volume VFH is calculated from the equation:

VFH= (o,K,, (K2, + T- Tgl) + co2KI2 (K22 + T- Tg2) (7-16)

where Tgt, Tg2 are the glass transition temperatures of 1 and 2, and Klh K2i, KI2and K22 are free-volume parameters of 1 and 2, determined experimentally.

The diffusivity (D = £>;) of trace amounts of a solute (1) in a glassy polymer(2) is given by the simplified equations:

D, = Do exp(-E/RT) exp [ -(yco2 £ V'2) I VFm ] (7-17)

and

Tg2)} (7-18)

where /L= 1 - (a2-a2g), and a2wd a2g are the thermal expansion coefficients ofthe rubbery and glassy states of the polymer.

The free-volume theory predicts the following changes of diffusion coeffi-cient (Duda and Zielinski, 1996): Strong effect of temperature and concentration

250 Chapter 7

near the glass transition temperature; increase with the size of solute molecule;plasticizers increase the available free volume, decrease the Tg, and increase thediffusivity; addition of impermeable fillers reduces D by increasing the tortuosityof the diffusing solute.

Yildiz and Kokini (1999) modified the free-volume theory to account forthe effect of temperature and water activity on the retention and release of flavorcompounds in food polymers. The diffusivity of hexanol, hexanal, and octanoicacid in uncooked soy flour was predicted to decrease sharply as the temperature isreduced in the rubbery state until the Tg, leveling-off at lower temperatures (glassystate). The diffusivity of flavor compounds in gliadin was predicted to increasesharply from about 1 x 10~18 m2/s to 1 x 10~10 m2/s, as the water activity was in-creased from 0.2 to 0.8 (at 25°C). Cross-linking of food polymers, e.g. by cookingof soy flour, predicts significant increase of diffusivity (i.e. reduced retention) offlavor compounds (e.g. hexanal).

3. Molecular SimulationMolecular simulations can describe sorption and diffusion phenomena in

polymer systems, based on chemical constitution of the components. Most of thesimulation work is related to simple amorphous rubbery and glassy systems, inwhich solute transport is assumed to follow the solution-Fickian diffusion mecha-nism of mass transport (Theodorou, 1996).

Molecular simulations are essentially solutions of the statistical mechanicsof a model of given molecular geometry and interaction parameters. They involvethe generation of configurations of the system, from which structural, thermody-namic and transport properties can be extracted. Molecular dynamics (MD) as-sumes that the penetrant (solute) moves into channels of the sorption sites, createdby small fluctuations in the polymer configuration.

Transition state theory (TST) provides a more approximate treatment of thepenetrant diffusion process, assuming a jumplike transport mechanism. The com-puter time required for the extensive computations can be reduced by certain ap-proximations, which are less severe than the ones used in the dual-sorption andfree-volume models. Computer calculations involve the estimation of the Henryconstant, the geometric characteristics of the accessible volume in the polymermatrix, and its distribution and rearrangement with thermal action, using MonteCarlo algorithms.

Molecular dynamics simulations have successfully predicted the self-diffusion coefficient in glassy and rubbery polymers, interacting with penetrantsolutes. The objective of molecular simulations is to develop the field of applied"molecular engineering of materials" for producing materials with tailored separa-tion and barrier properties.


IV. DIFFUSION OF SOLUTES IN FOODS

The diffusivity of solutes and other molecules in food materials dependsprimarily on the size of the diffusing molecule and the food structure. The neededexperimental measurements of diffusivity in solid and semisolid foods are usuallybased on the concentration-distribution method, described in Chapter 5 (Naessenset al., 1981, 1982; Giannakopoulos and Guilbert, 1986). Diffusivity data on salts,organic and flavor components are of particular interest to food processing andfood quality.

A. Diffusivity of SaltsTable 7.4 shows typical diffusivities of sodium chloride in model food gels

and food materials. The diffusivity depends strongly on the physical structure ofthe food material.

The diffusivity D of salt in dilute gels (Gros and Ruegg, 1987) is very closeto the D of salt in aqueous solutions, i.e. 12.5 * 10"10 m2 / s (see Table 2.4). Similarhigh diffusivities are observed in high-moisture foods of gel structure, like pickles(Pflug et al., 1975). Evidently, the salt ions can migrate in such gels at rates similarto the diffusion in liquid water.

The salt diffusivity in Swiss cheese (Gros and Ruegg, 1987) is considerablylower than in gels (1.9 x 10"10 m2/s), evidently due to the higher solids concentra-tion and the presence of fat globules in the material. Higher salt diffusivity valuesD were reported by Pajonk et al. (2000) in brining Swiss cheese. The D value de-creased from about 7 x 10"10 to 2 x 10"10m2/s when the brine concentration wasincreased from 0 to 20% NaCl. The diffusivity of salt in white feta cheese wasdetermined as 2.3 x 10"10 m2/s (Yanniotis et al., 1994).

Table 7.4 Diffusivities D of Sodium ChlorideFood Materials (20°C)

MaterialAgar gel, 3 % solidsPicklesSwiss cheeseMeat muscle, freshMeat muscle, thawedHerringGreen olives, freshGreen olives, treated

D, x 10"'°12.011.01.902.204.002.300.381.95

in

m2/s

252 Chapter 7

The salt diffusivity in fresh meat muscle is 2.2 x 10"'° m2/s, while it is con-siderably higher (4.0 x 10~10 m2/s) in meat flesh that has been frozen and thenthawed (Dussap and Gros, 1980; Fox, 1980). The relatively low D of salt in themeat is caused by the resistance of the cellular structure to mass transfer. The saltdiffusivity in fish is, in general, similar to the D in meat, e.g. 2.3 x 10"10 m2/s inherring (Rodger et al, 1984).

The diffusivity of salt in fresh green olives is quite low (0.38 x 10"'° m2/s),evidently due to the presence of skin and to high oil concentration. Treatment ofthe olives with 1.8% caustic soda increases the D value to 1.95 x 10"10m2/s (Dru-sasetal., 1988).

The diffusivity of sodium hydroxide in tomato skin, measured with a modi-fied diffusion cell (Figure 7.1), was found to be 0.02 x 10"10m2/s (Floras et al.,1989). A higher value was found for the diffusivity of the same alkali in the skinof pimiento pepper (0.055 x 10~10m2/s).

Diffusivities of other salts of interest to foods (chlorides, nitrites, nitrates,etc.) are similar to the D values of sodium chloride. A bibliography on the diffu-sivity of salt in foods was prepard by Ruegg and Schar (1985).

B. Diffusivity of Organic ComponentsThe diffusivity of organic solutes in food materials is important in food

processing operations, like extraction (sugars, lipids, flavors), and in food quality(e.g. sugar taste, volatile flavor retention).

The diffusivity D of organics in liquid foods is related closely to the viscos-ity 77 of the solution, through the relation r/D/T= constant Eq. (2-36). Organolepticflavor perception is related to the diffusivity of the flavor component (e.g. sugar)and the viscosity of the liquid food (Kokini et al., 1982; Kokini, 1987). The flavorof highly viscous pseudoplastic foods is enhanced by shearing, which reducesconsiderably the apparent viscosity, increasing at the same time the diffusivity ofthe flavor component(s).

For large molecules in food liquids, like peroxydase, the Stokes-Einsteinequation (7-6) can be applied, while for smaller solutes in sugar solutions (e.g.nicotidamine) the Wilke-Chang equation (2-34) has been found applicable (Lon-cin, 1980; Stahl and Loncin, 1979).

The diffusivity of nicotinamide in fructose solutions decreases from about8 x 10~10 to 0.5 x 10~10 m2/s, when the sugar concentration is increased from 0 to60%. In the same range of fructose concentration, the diffusivity of peroxidasedecreases from 1.0 x 10"10 to 0.1 x IQ"10 m2/s. The activation energy for diffusionof both species increases sharply from 20 to 45 kJ/mol in the same sugar concen-tration range.

The prediction models for the diffusivity of solutes in polymers, discussedearlier in this chapter, are difficult to apply in solid and semisolid foods, duemainly to the heterogeneous physical structure of the food materials. The presence


of significant open space in food solids, such as pores, cracks, and channels, com-plicates the diffusion process, since a portion of the solutes can diffuse quickly inthe gas phase, while the rest diffuses very slowly from the sorbed or trapped state.The diffusivity in the gas phase is about five orders of magnitude (x 105) higherthan in the solid phase.

The free-volume model, suggested for the prediction of diffusivities inpolymers, was applied by Yildiz and Kokini (1999) for the prediction of diffusiv-ity of flavor components in solid foods. Application of this model assumes that thefood material behaves as a homogeneous polymer material of low porosity, suchas uniform protein, carbohydrate or lipid films.

The molecular simulation model (Theodorou, 1996), requiring extensivecomputer calculations, when developed and applied further in the polymer field,could be adapted to food materials in the future.

Table 7.5 shows some typical diffusivities of organic solutes in food materi-als, which are useful in calculations involving solvent extraction (leaching) andliquid infusion operations (Schwartzberg and Chao, 1982). The diffusivity of sug-ars in gels (e.g. agar) is similar to the diffusivity in water solutions, Table 7.3(Warin et al., 1997).

The diffusivity of solutes in solid foods D is considerably lower than in di-lute water solutions, shown in Tables 7.2 and 7.3, due to blockage of diffusionpaths, occlusion (trapping), and sorption by the food biopolymers. The D in solidsis related to the diffusivity of the solutes in water Dw by an empirical relationanalogous to Eq. (5-2):

D = (ew/r)Dw (7-19)

where ew is the volume fraction of free water in the solid (analogous to porosity),and T is the tortuosity of the diffusion path.

Table 7.5 Diffusivities of Solutes in Solid FoodsSolidSugar beetsSugar caneApple slicesCoffee beansSoybean flakesCottonseed oilPeanuts

SoluteSucroseSucroseSugarsCoffee solublesSoybean oilCottonseed oilPeanut oil

SolventWaterWaterWaterWaterHexaneHexaneHexane

r,°c65757598696925

D, x 10'10

6.802.0011.51.001.000.270.006

7,m /s

254 Chapter 7

The free water fraction in the solid can be estimated from the moisture con-tent and the sorption isotherm, but the tortuosity factor must be estimated indi-rectly from the measured D. Both parameters are not constant during food process-ing and storage, due to the significant changes of the food structure.

The effect of solids content on the diffusivity of organic compounds infoods, is illustrated by the diffusivity of cyclohexanol in potato, which decreasesfrom 6 x 10'10 to 2 x 10~10 m2/s in high solids potato (Loncin, 1980). The activationenergy for diffusion is analogous to that of water in potato, 35.7 kJ/mol.

The diffusivity of a solute may be reduced significantly by the presence ofanother solute, diffusing simultaneously in a solid food material (multicomponentdiffusion). Thus, the individual diffusivity of citric acid (1) in prepeeled potato isreduced from /)/ = 4.3 x 10~10 to D12 = 6.6 x 10"" m2/s in the presence of ascorbicacid (2), diffusing simultaneously. At the same time, the diffusivity of ascorbicacid is reduced from D2 = 5.4 x IQ'10 to D2, = 8.3 x IQ"11 m2/s (Lombardi et al,1996). The diffusivities of the two solutes in dilute water solutions (w) are D!w =6.6 x 10'10andZ)2w= 8.4 x I(r10m2/s.

C. Volatile Flavor RetentionThe diffusion of volatile flavor (aroma) components in foods is important in

food processing operations, such as evaporation and drying, and in storage andquality of food products. Most aroma components are very volatile in aqueoussolutions, since they form highly nonideal mixtures with water. The volatility ofthese components at thermodynamic equilibrium is characterized by the activitycoefficient and the relative volatility, which are the basic elements of the vapor-liquid equilibria (VLB). Calculation of (VLB) is required for the analysis of anyvapor-liquid separation or interaction (Prausnitz et al., 1986; Reid et al., 1987; LeMaguer, 1992).

The relative volatility of an aroma compound A in dilute water solution aAwis defined by the equation (Saravacos, 1995)

aAw=yApAo/Pwo (7-20)

where yA is the activity coefficient of A, and pAO, pHO are the vapor pressures of Aand water, respectively, at the given temperature.

The activity coefficient of a component YA is related to the concentration Qand the chemical activity aA by the equation:

aA = /ACA (7-21)


The activity coefficients of aroma components in water and aqueous foodsare very high, especially for partially soluble organic flavor components, like es-ters and higher alcohols. They are estimated by computer-aided techniques usingempirical models, like the UNIQUAC and the UNIFAC (Reid et al., 1987). Thepresence of sugars in aqueous solutions, like in food materials, increases consid-erably the activity coefficient (Saravacos et al., 1990; Sancho and Rao, 1997).

Table 7.6 shows some typical relative volatilities of volatile flavor com-pounds in dilute water solutions (Saravacos, 1995; Chandraskaren and King,1972).

The relative volatility of these compounds in aqueous solutions of 60% su-crose is 20 to 10 times higher than in water, due to the strong interactions of the 3-component system (Saravacos et al., 1990).

The volatile flavors (aromas) are normally recovered during the evaporationof fruit juices and other aqueous systems by stripping and distillation processes(Saravacos, 1995; Karlsson and Tragardh, 1997). Maximum removal of a volatilefrom the liquid phase is obtained when vapor-liquid equilibrium (VLB) is estab-lished. However, establishment of true equilibrium requires infinite time, soevaporation and distillation are actually nonequilibrium processes with partialremoval of volatiles. Diffusion of the flavor components from the interior to thesurface of food particles is reduced sharply in the presence of sugars and othersolids. Evaporation from falling liquid films (Lazarides et al., 1990) or from me-chanically agitated films (Marinos-Kouris and Saravacos, 1974) can increase thestripping efficiency of volatiles.

Table 7.6 Typical Relative Volatilitiesof Aroma Compounds in AqueousSolutions aAw at Infinite Dilution (25°C)

Volatile compound aAw

Methyl anthranilateMethanolEthanol1 -Propanol1-Butanoln-Amyl alcoholHexanol2-ButanoneDiethyl ketoneEthyl acetateEthyl butyrate

3.908.308.609.5014.123.031.076.077.0205643

256 Chapter 7

The retention of volatile flavors during food dehydration depends primarilyon the presence of sugars and other solids, which reduce the aroma diffusivity inthe food material. Contrary to aroma recovery processes, aroma retention is ahighly nonequilibrium process, utilizing conditions that will prevent the flavorcompounds from reaching the evaporation surface, such as fast surface drying(Rulkens and Thijssen, 1972).

The loss of volatile flavors depends on the evaporation or drying rate of wa-ter. The mass transport of volatiles should be considered as a ternary diffusionprocess, with three binary diffusivities, i.e. water/solids, volatile/water and vola-tile/solids (Coumans et al., 1994a).

Figure 7.4 shows the loss of a very volatile flavor compound, ethyl butyrate(relative volatility in water aAw = 643), as function of % water evaporated inaqueous solutions and during vacuum- or freeze-drying (Saravacos and Moyer,1968a, b). The loss of the volatile ester from the water solution is very rapid, e.g.90% loss by evaporation of 30% water. The presence of pectin in the water solu-tion reduces the volatile loss and increases its retention. A higher retention is ob-tained by freeze-drying.

Volatile retention during spray drying depends not only on the relative vola-tility but also on the interaction of the compound with the nonvolatile componentsof the food liquid. Thus, in spray drying of food emulsions containing flavors,ethyl butyrate is retained only by 20%, while limonene may be retained almostquantitatively (Furuta et al., 2000).

The retention of volatile flavors during food dehydration is a very importantconsideration in the selection of drying processes and equipment for optimumproduct quality. Flavor retention is related to the reduction of diffusivity of flavorcompounds by sugars and other food solids. Figure 7.5 shows that the diffusivityof diacetyl in water solutions is reduced by almost 100 times, when the sugar con-centration is increased from 0 to 70% (Voilley and Simatos, 1980; Voilley andRoques, 1987).


50

Evaporation (%)

100

Figure 7.4 Retention of ethyl butyrate: W, evaporation of water; VD, vacuum-drying ofpectin solution; FD, freeze-drying of pectin solution.

Thermodynamic and transport phenomena analysis indicate that flavor retention isa diffusion-controlled process (Kerkhof, 1975; Bruin and Luyben, 1980). Fastdrying processes, like spray drying, improve volatile retention by trapping thesolute in the solid matrix. A selective diffusion mechanism may explain the vola-tile retention in spray- and freeze-drying (Coumans et al., 1994b). Atomizationand evaporation of water/volatiles from drops control flavor retention in spray-dried particles (King, 1994; Hecht and King, 2000).

Retention of charactertistic aroma during storage of dried fruits can be im-proved by using low relative humidities (Rff) and low temperatures. Moisturesorption of stored fruits increases sharply at RH > 60%, resulting in a strong rise offlavor diffusivity and subsequent loss of aroma (Saravacos et al., 1988).

258 Chapter 7

l.E-09

l.E-10

l.E-11

\

l.E-1220 40

Sugar(%)

80

Figure 7.5 Diffusivity of diacetyl in sucrose solutions.

D. Flavor EncapsulationEncapsulation and controlled release of solutes is used widely in pharma-

ceuticals, medicinal products, flavors, and pesticides. Controlled release is basedon relaxation-controlled dissolution of the coating material, which consists usuallyof a glassy polymer (Cussler, 1997).

Encapsulation of flavors, acidulants (citric and ascorbic acid), salts, and en-zymes is used to prevent or control the diffusion of the solutes in various foodprocessing and food utilization operations (Karel, 1990). Encapsulation can beachieved by entrapment in glassy polymers or in sugar crystals, in fat-based matri-ces, or by incorporation in liposomes (e.g. lecithin).

Release of encapsulated solutes is achieved by temperture and moisture con-trol, enzymatic release, grinding etc. The role of glass transition temperature Tg tosolute release is important, since diffusivity rises sharply above Tg. The WLFequation (7-10) relates the diffusivitiy to the temperature and the Ts. The "collapse


temperature" is related to T& and both temperatures decrease as the moisture con-tent is increased.

Spray- and freeze-drying are used to encapsulate flavor solutes in polymermatrices, using high initial drying rates to form a dried polymer layer, which re-duces diffusivity.

V. PERMEABILITY IN FOOD SYSTEMS

The transport of small solutes, such as water, oxygen, and carbon dioxidethrough polymer films and protective coatings is of fundamental importance tofood packaging and food processing. The permeability of these materials is basedon the principles of diffusion of solutes in polymer systems. The permeability ofsynthetic membranes is important to separation processes used in food processing,such as reverse osmosis, gas separation, and ultrafiltration.

The structural and physicochemical factors, which affect the diffusivity ofsolutes in polymers, are also important in characterizing the performance of pack-aging films and food coatings. Control of such factors as glassy/rubbery state,cross-linking, and polymer orientation, can determine the permeability of thesematerials.

A. PermeabilityThe permeability P of a film or thin layer of thickness z is related to the dif-

fusivity D and the solubility S of the penetrant (solute) in the material, accordingto the equation:

J = P (Aplz) = DS (Aplz) (7-22)

where J is the mass transfer rate (kg/m2s), Aplz is the pressure gradient (Pa/m), and5 is the gas/liquid equilibrium constant, S = C/p where C is the concentration(kg/m3) and p the pressure (Pa). The solubility S is equal to the inverse of theHenry constant (S=1/H), and it has units (kg/m3Pa); it can be determined as theslope of the sorption isotherm (C versus p).

From equations (7-22) it follows that:

P = DS (7-23)

The permeability has SI units (kg/m s Pa) or (g/m s Pa), but various otherunits are used in packaging, reflecting the measuring technique or the particularfood/package application (Hernandez, 1997; Donhowe and Fennema, 1994).

260 Chapter 7

The permeability P is related to the permeance PM or transmission rate 77?(=PM) and the water vapor transfer rate WVTR by the equation (McHugh andKrochta, 1994):

P = PMz = WVTR I Ap (7-24)

The units of permeance (kg/m2 s Pa) are identical to the units of the mass transfercoefficient kp. The units of WVTR are (kg/m s) (Saravacos, 1997).

The SI units are useful in relating and comparing the literature data on P andWVTR to the fundamental mass transport property of diffusivity D (m2/s).

The permeability of polymer films and coatings can be determined bymeasurements of sorption kinetics and diffusion, discussed in Chapter 5, in rela-tion to water transport. Conversion of solubility S and diffusivity D data to perme-ability P though Eq. (7-23) is possible, when the material behaves like a homoge-neous medium and Fickian diffusion can be assumed.

Simplified permeability measurement methods are used for packaging andcoating films (barriers), and most of the literature data are reported in units relatedto the special methods used (ASTM, 1990, 1994). The measured permeabilitiesrepresent an overall transport property of the material, based on the applied pres-sure gradient. Since the polymer film may have structural inhomogeneities, suchas pores, channels, cracks, and pinholes, mass transport may involve, in additionto molecular diffusion, Knudsen diffusion and hydrodynamic or capillary flow(Hernardez, 1997). In such cases, the simplified relationship between diffusivityand permeability Eq. (7-23) is not applicable.

Permeability is affected significantly by environmental condition, such as airrelative humidity (RH), which may increase sharply the permeability of mostpackaging and coating films.

The total permeability PT of a multilayer laminate is related to the perme-abilities and the thicknesses of the individual films (P, z/) by the equation (Cook-seyetal., 1999)

/V=[(Sr,)]/[W/)] (7-25)

The total permeance PMT or transmission rate TRT can be calculated from theequation:

P M T = l / I , ( z , / P d (7-26)

Temperature increases permeability P according to the Arrhenius equationin a similar manner with the effect of temperature on diffusivity D and solubilityS:

= P0 exp(-£//?7), D = D0 exp(-ED/RT), S = S0 exp(-Es /RT) (7-27)


Table 7.7 Conversion Factors to SI Permeability Units (g/m s Pa)Conversion from / to (g / m s Pa) Multiplying factorcm3(STP) mil /100 in2 day atm 6.42 x 10'17

cm3(STP) mil / m2 day atm 4.14* 10"18

cm3(STP) urn / m2 day kPa 1.65 x 1Q'17

g)im/m2daykPa 1.16X10' 1 4

gmm/m2daykPa 1.16x10""g mil/m2 day atm 2.90 x 10"15

g mil/m2 day (mm Hg) 2.20 x 10'12

g mil/m2 day (90 % RH, 100 °F) 4.50 x 10'14

g mil/100 in2 day (90 % RH, 100 °F) 7.00 x 10'13

perm (ASTM, 1990)____________________1.45 x 1Q-9

STP = standard temperature and pressure. 1 mil = 0.001 inch = 2.54xl00m.Pressure drop of water vapor across the film at 90/0% RH and 100°F,AP = 6560 Pa. 1 mm Hg = 133.3 Pa.

The energy of activation for permeability Ep may vary, depending on thetype of polymer and the temperature, in the wide range of 10 to 80 kJ/mol (Her-nandez, 1997).

Table 7.7 shows the conversion factors from the various literature units ofpermeability to SI units (g/m s Pa).

B. Food Packaging FilmsSynthetic polymer firms are used as barriers to the transport of water vapor,

oxygen, carbon dioxide, and food components, like aroma/flavor compounds andlipids, from or to the packaged food product. Food packaging films are made ofspecial polymeric materials, like polyethylene, both low density (LDPE) and highdensity (HDPE), polypropylene (PP), polyvinyl chloride (PVC), polystyrene (PS),polyethylene terephthalate (PET) and polyamides (nylon) (Hernandez, 1997;Miltz, 1992). Permeability, mechanical properties, food compatibility (non-toxicity), and cost are the main characteristics in selecting the proper material(Brody and Marsh, 1997; Hanlon et al., 1998). Permeability, like diffusivity, isaffected significantly by polymer microstructure, solute-polymer interactions, sol-ute concentration (especially moisture content or RH), and temperature.

Some typical permeabilities of common packaging films to water vapor andoxygen are shown in Table 7.8 at 25°C (Miltz, 1992; Hernandez, 1997).

262 Chapter 7

Table 7.8 Permeabilities of Packaging Films to WaterVapor and Oxygen (25°C)

Packaging filmLDPEHDPEPPPVCPSPETNylon

Permeability, xWater vapor

1.400.201.003.0012.01.400.002

10- 1 2 g/msPaOxygen

0.0310.0070.0100.0050.0180.00060.0004

LDPE = low density polyethylene, HOPE = high density polyethylene,PP = polypropylene, PVC = polyvinyl chloride, PS = polystyrene,PET = polyethylene terephthalate, Nylon = polyamide.

The permeability of nylon to oxygen at various moisture contents has beenanalyzed by the dual-sorption model (Hernandez, 1994). Although the water diffu-sivity increases at higher moistures, the solubility and the permeability decreasesharply at the beginning, leveling-off at water activities above 0.2.

The permeability of polymer firms and food coatings to carbon dioxide isimportant in food packaging and storage. Typical values of permeability of carbondioxide at 25°C are: LDPE, 1.6 x 10'13; HDPE, 5.3 x 10'15 g/m s Pa.

C. Food CoatingsThe permeability of edible food coatings to water is of particular interest to

food quality, since their primary function is to act as barriers to moisture transportduring storage. Food coatings can also control the transport of gases (mostly oxy-gen), flavor components and lipids in food systems. Edible coatings, used as barri-ers in foods, include proteins (wheat gluten, caseinates, whey protein, corn zein),polysaccharides (starch, dextrins), pectins, lipids, and chocolate. Composite coat-ings, containing a food biopolymer (e.g. protein) and a hydrophobic material, likelipid, fatty acid, chocolate, and beeswax, usually have very low water permeabil-ities.

The food coatings are prepared as solutions or dispersions/emulsions of theprimary biopolymer in solvents (ethanol, alkalis, or acids). They contain variousplasticizers, such as glycerine and sorbitol, which improve the physical and me-chanical properties of the coating. They are applied to the various fresh and proc-essed foods, like fruits and vegetables by dipping in an emulsion, spraying orfoaming and brushing.

Table 7.9 shows typical water permeabilities of food coatings (McHugh andKrochta, 1994):


Table 7.9 Water Vapor Permeabilities Pof Food Coatings (25°C)

Food coatingGluten-glycerineWhey protein-sorbitolZein-glycerineSodium casemateChocolateBeeswax

P, x lO- 1 0 g/msPa6.107.201.004.200.120.006

D. Permeability/Diffusivity RelationThe simple permeability/diffusivity/solubility relation of Eq. (7-23) is useful

for estimating the permeability P from diffusivity D and solubility S data of poly-mer films, and for comparison of P and D data. This relation applies to systemsbehaving as homogeneous materials, in which solute transport is by Fickian mo-lecular diffusion. It does not hold for heterogeneous materials, consisting of pores,channels and capillaries, in which a significant portion of mass transfer takes placeby mechanisms other than molecular diffusion. Table 7.10 shows some typicaldiffusivity and permeability data for packaging films and food coatings. The com-parison is facilitated by using consistent (SI) units (Saravacos, 2000).

A typical application of the permeability-diffusivity relation is given forchocolate film, using published data of Biquet and Labuza (1988): Typical perme-ability P and diffusivity D values for a chocolate coating about 0.6 mm thick at200C:P = 0.11 x 10- '°g/msPaandZ)=l x 10-13m2/s. The solubility S of water inthe chocolate material can be estimated from the sorption isotherm at 20°C. It isdefined by the Henry equation, C = S p, where C is the concentration (kg/m3) inthe material andp is the partial pressure of water (Pa). Thus, the solubility is equalto the slope of the isotherm (S = C/p). Considering the initial sorption stage, wateractivity a,v 0 to 0.1, S = (1.7 kg water/100 kg solids)/Ap where Ap = a,vp0 or Ap =0.1 PO, and PO is the vapor pressure of water at 20°C (p0 = 2340 Pa), and Ap =234Pa. The concentration of water in the chocolate material is converted to consis-tent (SI) units, as follows: Assume density of dry chocolate 1600 kg/m3; therefore,the volume of 100 kg dry material will be 100/1600 = 0.0625 m3. The water con-centration in the chocolate becomes C = (1.7/0.0625) = 27.2 kg/m3, and the solu-bility S = 27.2/232= 0.116 kg/m3 Pa.

Using the measured diffusivity of the system (D = 1 x 10~13 m2/s), the per-meability of the chocolate film according to Eq. (7-23) will be P = D S = 0.116 x10"13kg/ms Pa, or P = 0.116 x 10"'°g/ms Pa, which is very close to the measuredpermeability.

264 Chapter 7

Table 7.10 Typical Water Vapor Permeabilities and DiffusivitiesFilm or coatingHDPELDPEPPPVCCellophaneProtein filmsPolysaccharide filmsLipid filmsChocolateGlutenCom pericarp

P, x 10-'°g/msPa0.0020.0140.0100.0413.70

0.10-10.00.10-1.00

0.003-0.1000.115.001.60

A xio- '°m2 /s0.0050.0100.0100.0501.00

0.1000.1000.0100.0011.000.10

LPDE = low density polyethylene, HDPE = high density polyethylene,PP = polypropylene, PVC = polyvinyl chloride

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Hernandez, RJ. 1994. Effect of Water Vapor on the Transport Properties of Oxy-gen through Polyamide Packaging Materials. J. Food Eng. 22:509-532.

Karel. M. 1990. Encapsulation and Controlled Release of Food Components. In:Biotechnology and Food Process Engineering. H.G. Schwartzberg and M.A.Rao, eds. New York: Marcel Dekker, pp. 277-293.

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268 Chapter 7

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8Thermal Conductivity and Diffusivityof Foods

I. INTRODUCTION

The thermal transport properties, thermal conductivity and thermal diffusiv-ity of simple gases and liquids can be predicted by molecular dynamics and semi-empirical correlations, and numerous tables and data banks are available in theliterature (Chapter 2). Experimental measurements are necessary for the thermaltransport properties of foods, due to their complex physical structure. Empiricalmodels have been proposed for the correlation of experimental data and the possi-ble explanation of the heat transport mechanisms.

The thermal conductivity (X) of a material is a measure of its ability to con-duct heat and is defined by the basic transport equation (2-3), which is integratedto give:

q/A=A(TrT2)/x (8-1)

where qlA is the heat flux (W/m), x is the thickness of the material (m), T, and T2are the two surface temperatures of the material, and A is the surface of the mate-rial normal to the direction of heat flow (m2). The S.I. units of A are W/mK. Equa-tion (8-1) is the basis for the direct measurement of A (guarded hot-plate method).

The thermal diffusivity a of a material can be estimated from the thermalconductivity A using the equation:

a = JJpCp (8-2)

where p is the density (kg/m3) and Cp is the specific heat (J/kgK) of the material.The S.I. units of a are m2/s.

269

270 Chapter 8

The thermal conductivity of foods depends of the chemical composition, thephysical structure, the moisture content, and the temperature of the material. The Aof unfrozen foods varies between the A, of air (0.020 W/mK) and water (0.62W/mK). Higher A values characterize the frozen foods (about 1.5 W/mK).

The thermal diffusivity of foods does not change substantially, because anychanges of A are compensated by changes of the density of the material Eq. (8-2).Typical values of a for unfrozen food are 1.3xlO"7 m2/s and for frozen food 4xlO"7

m2/s.The thermal conductivity of solid foods is a strong function of the porosity

of the material. This variation is about one order of magnitude, compared to thevery wide variation of mass diffusivity in porous foods. The changes in heat andmass transport properties of porous foods reflect the differences in /I and D ofgases and liquids, according to the approximate relations:

A(gas)//l(liquid)=l/10 (8-3)

£>(gas)/£>(liquid) = 10000/1 (8-4)

Empirical models of thermal conductivity, analogous to the models of elec-trical conductivity, can be used to correlate the experimental data. The literaturedata on X can be analyzed statistically, using correlations analogous to the modelsof moisture diffusivity (see Chapter 6).

II. MEASUREMENT OF THERMAL CONDUCTIVITY AND DIFFUSIVITY

The measurement of the thermal transport properties of foods is describedby several authors in the literature, notably by Mohsenin (1980), Nesvadba (1982),Sweat (1995), Rahman (1995), and Urbicain and Lozano (1997). A comprehen-sive study of the subject was undertaken within the collaborative research projectCOST 90 in the European Union (Meffert, 1983; Kent et al., 1984).

A. Thermal ConductivityTwo experimental methods are normally used for the measurement of the

thermal conductivity (X), i.e. the guarded hotplate and the heated probe. Othermethods, suggested for food materials, are the Fitch method and its modifications,the thermal comparator method, and the temperature history method (Rahman,1995).

Thermal Conductivity and Diffusivity of Foods 271

1. Guarded Hot PlateThe guarded hot-plate method is based on the Fourier equation for steady-

state heat flux (8-1). The experimental apparatus is diagrammatically shown inFigure 8.1. Details of the apparatus are given by Drouzas and Saravacos (1985).

The apparatus consists of two circular brass plates, between which the sam-ple material is placed. The upper plate is heated electrically and the lower coldplate is maintained at a constant temperature. Unidirectional flow of heat is as-sured by two guard rings around the plates. After establishment of steady state, theheat flow is measured with an electrical meter and the thermal conductivity (X) isdetermined from equation (8-1).

Although the guarded hot plate is an accurate method, it requires specialprecautions, like uniform sample thickness, good contact with the plates, and rela-tively long time to reach steady state, which may change the moisture content ofthe material.

2. Heated ProbeThe heated probe method is faster and it requires less sample material. For

these reasons it is used more widely than the guarded hot-plate method. The hotprobe is a transient method, based on the measurement of the sample temperatureas a function of time, while the sample is heated by a known line heat source. As-suming that the line heat source is an infinite medium, and that the heat flow isradial, the temperature T at a point very close to the line source, after a time t willbe (Rahman, 1995):

quard rings

81_

••f'H.4.-^ *-:; -a.- :-; "'. s-y "£• ' %-'•-"-'"- •<<.':£ ":-

AAAAAA^

iffigam

Jjpyjij*

hot plate

/ , sample/

cold plate

Figure 8.1 Diagram of a guarded hot plate apparatus.

272 Chapter 8

(8-5)

where ft is a constant, and q is the heat flow per unit length of the line source(W/m).

The temperature difference (ArP=T2-T1) after times t} and tt of heating willbe:

Thus the thermal conductivity A can be obtained from the slope (qttnX) of a se-milog plot of T versus t.

The heated probe method requires short measurement times (less than aminute), and relatively small samples.

heated wire

seal

seal

thermocouple

stainlesssteelneedle

fillingmaterial

Figure 8.2 Principle of heated probe.


Figure 8.2 shows schematically a heated probe used for thermal conductivity(Drouzas and Saravacos, 1988; Sweat, 1995). A stainless steel needle (or syringe)is used to house the line source (a heated constantan wire) and the thermocouple(chromel-constantan). Time-temperature data of the probe, inserted in the sample,are used to calculate /I from Eq. (8-7), which is derived directly from (8-6):

A = [23I2R logfo/?;)]/[4^)7] (8-7)

where / is the electrical current (A) and R' is the specific electrical resistance ofthe heating wire (H/m). The heated probe is calibrated with glycerin solutions orwith known insulating materials.

B. Thermal DiffusivityThe thermal diffusivity a is usually estimated indirectly from the thermal

conductivity A, the density p and the specific heat Cp of the material, according toEq. (8-2).

Direct measurement of a can be made using various methods, discussed byRahman (1995). All methods are based to some simplified solutions of the un-steady-state heat conduction equation (Fourier):

(8-8)

/. Dickerson MethodThe simplest experimental method is the transient method of Dickerson

(1965), modified by Poulsen (1982), and used by Drouzas and Saravacos (1985).The principle of the method is to obtain time-temperature data in a cylindricalsample of material, as shown in the diagram of Figure 8.3.

The transient heating time t in the cylindrical container is given by the semi-empirical equation (Ball and Olson, 1957):

t=f\Qg\j(Ts-T0)l(Ts-T)} (8-9)

where T0 is the initial temperature, Ts is the outside surface temperature, and T isthe temperature after time t. The linear semilog relationship is defined by the lagfactory and the reciprocal of the slope/ Equation (8-9) is used in thermal process(sterilization) calculations of the formula method.

274 Chapter 8

thermocouples

constanttemperature"

sample

Figure 8.3 Transient method for thermal diffusivity.

For a long cylindrical container with high surface heat transfer coefficientsh, i.e. for Biot numbers Bi > 40 (Bi = MX), the parameter/of the heating source isgiven by the equation:

/= 0.398(r2/a)

where r is the radius of the container.Thus, the thermal diffUsivity (a) can be obtained from Eq. (8-10).

(8-10)

2, Modified Heated ProbeThe thermal diffusivity (a) can be obtained experimentally, using a modified

heated probe (Mohsenin, 1980; Drouzas et al., 1991). The second thermocouple isinserted in the basic probe (Figure 8.2) at a fixed short distance r from the initialthermocouple. The temperature rise AT at the distance r at the heating time t isgiven by the following solution of the unsteady-state heat conduction in a cylindri-cal solid:

AT= (q/4aX) [-0.58/2 -

where j3 = r/2(af)1'2! - (34/4x2! (8-11)


Thus, a can be calculated from Eq. (8-11), using the corresponding A valuefrom a parallel measurement with the normal probe.

The exact distance r of the two thermocouples is critical in the modifiedprobe method and a calibration of the system may be required. A comparison ofthe direct probe method with the indirect calculation of a indicated a better accu-racy of the indirect method (Drouzas et al., 1991).

III. THERMAL CONDUCTIVITY AND DIFFUSIVITY DATA OF FOODS

The literature contains several data on the thermal transport properties offoods in the form of tables, empirical models, regression equations, and databanks.

A. Unfrozen FoodsTable 8.1 shows typical values of thermal conductivity of foods: Kostaro-

poulos (1971), Mohsenin (1980), Rahman (1995), Sweat (1995), and Singh(1995). Sections IV and V of this chapter contain some mathematical models andstatistical compilations of thermal conductivity and diffusivity.

Table 8.1 Thermal Conductivity K of Unfrozen Foods (25°C)Food materialWaterSucrose solutionGranular starchGelatinized starchPotatoAppleTomato pasteBeefFishMilkVegetable oilDried fruitFreeze-dried gel

% Moisture100851585828565748085

184

A, W/mK0.6200.5670.1100.5470.5330.5130.5500.4520.4600.5300.1800.2370.041

276 Chapter 8

The thermal conductivity A of foods decreases, in general, as the moisturecontent of the material is reduced. The A of dried food materials decreases sharplyas the porosity is increased, e.g. by freeze-drying or puffing.

The /I of fibrous dry foods is higher parallel than across the fiber (Chapter3). Thus, the /I of freeze-dried cellulose gum (fibrous structure) was 0.0627 W/mKcompared to 0.0410 W/mK of porous freeze-dried starch gel (Saravacos and Pils-worth, 1965). An analogous effect was noticed in the drying rates of the two mate-rials (see Chapter 5).

Figure 8.4 shows the effect of air pressure on the thermal conductivity of afreeze-dried gel (Saravacos and Pilsworth, 1965). The A decreases sharply from0.041 to about 0.010 W/mK as the air pressure is reduced from atmospheric tobelow 1 mbar. Similar effects of pressure was observed in freeze-dried fruits(Harper, 1962) and milk and orange pulp (Fito et al., 1984).

The thermal diffusivity (a) of unfrozen foods does not change appreciablywith the moisture and temperature, ranging from about l.OxlO'7 to 1.5xlO"7 m2/s.Detailed tables of (a) for various food materials are presented by Rahman (1995).

B. Frozen FoodsThe thermal conductivity of frozen foods is significantly higher than the A. of

unfrozen materials, due to the higher /I of ice. The /I of a frozen food of 85% mois-ture content decreases gradually from about 1.5 W/mK at -40°C to 0.5 W/mK at0°C, as shown in the diagram of Figure 8.5. A sharp drop of A is observed near0°C, due to the melting of ice.

An analogous variation of the thermal diffusivity a of the frozen foods is ob-served below the freezing point (Figure 8.6). The a of a typical frozen food de-creases from about 1x10'6 m2/s at -40°C to 1x10'7 m2/s above 0°C. The sharp dropnear the freezing point is due to the sharp increase of cp, which incorporates theheat of melting, near 0°C (Singh, 1995).

C. Analogy of Heat and Mass DiffusivityThe heat and mass transport analogy in gases (Chapter 2) has been observed

in porous foods at low moisture content (Kostaropoulos and Saravacos, 1997). Theanalogy is based on the transport mechanisms in the gas phase, which becomesimportant in highly porous materials, like granular, freeze-dried, and extrusion-cooked foods. As in the case of moisture diffusivity (Figure 5.24), the thermaldiffusivity of granular starch shows a maximum near 15% moisture, and it in-creases gradually at moistures higher than 35% (Figure 8.7).


0.050

0.0000.1 10 100

Pressure (mbar)

1000

Figure 8.4 Effect of air pressure on thermal conductivity of porous food material.

-40 -30 -20 -10 0 10 20 30 40

Temperature (°C)

Figure 8.5 Thermal conductivity of frozen food material.

278 Chapter 8

l.OE-06

-40 -30 -20 -10 0 10 20

Temperature (°C)

Figure 8.6 Thermal diffusivity of frozen food material.

30 40

10 20 30 40

Moisture Content (%)

50 60

Figure 8.7 Thermal diffusivity versus moisture content of porous food material.


D. Empirical RulesA preliminary checking of thermal transport property data can be made, us-

ing the empirical rules, suggested by Kostaropoulos (1981):

I. Thermal Conductivity

i. Food with moisture content (Xw) > 30-40%Xi~ 0.40- 0.58 W/mK

ii. Frozen foods, Xw > 30-40%X K ~ 2.5 A, j

iii. Dry food

iv. Fats and oilsX i v~ 0.25- 0.50

2. Thermal Diffusivity

i Foods, Xw>30%ctj^ 1.4xlO"7r

ii. Frozen food

iii. Dry food

iv Fats and oils

280 Chapter 8

IV. MODELING OF THERMAL TRANSPORT PROPERTIES

A. Composition ModelsSeveral composition models have been proposed in the literature, most of

which are summarized by Miles et al. (1983), Sweat (1995), and Rahman, (1995).The most promising seems to be the model proposed of Sweat (1995):

A = Q.5SXv + Q.l55Xp+Q.25Xc+Q.l6Xf+Q.135Xa (8-12)

where Xm Xp, Xf and Xa are the mass fractions of water, protein, fat, and ash, re-spectively.

The above model was fitted to more than 430 liquids and solid foods withsatisfactory results. It is not accurate for porous foods containing air, for whichstructural models are needed.

The thermal conductivity of water in the above equation was fitted to about0.58W/mK which is less than the thermal conductivity of pure water, 0.605W/mK.Either the selected data are biased, or they indicate that the effective thermal con-ductivity of water in foods is less than the thermal conductivity of pure water(Sweat, 1995).

The key to the accuracy of the above equation is having accurate values forthe thermal conductivity of "pure" components. This is easy for the water and oilfractions but very difficult for the other fractions. In fact, the thermal conductivityof proteins and carbohydrates probably varies according to their chemical andphysical form.

However, it is not needed to find more accurate additive composition mod-els, because of the inherent inaccuracy in the composition models, which theydon't take into account the geometry of the component mixing. As in the case ofair-containing foods, structural models must be used. The temperature effect is notincluded in the above equation. Thus, it is valid at the fitting region approximatelyat 25°C. The temperature effects of the major food components are summarized byRahman (1995) in Table 8.2 and in Figure 8.8.


Table 8.2 Effect of Temperature on the Major Food Components

X=bo+biT+b2T2+b3T:1

Air

Protein

Gelatin

Ovalbumin

Carbohydrate

Starch

Gelatinized Starch

Sucrose

Fat

Fiber

Ash

Water

Ice

bo

2.43E-02

1.79E-01

3.03E-01

2.68E-01

2.01E-01

8.7 IE-02

3.22E-01

3.04E-01

1.81E-01

1.83E-01

3.30E-01

5.70E-01

2.22E+00

b,

7.89E-05

1.20E-03

1.20E-03

2.50E-03

1.39E-03

9.36E-04

4.10E-04

9.93E-04

2.76E-03

1.25E-03

1.40E-03

1.78E-03

-6.25E-03

b2

-1.79E-08

-2.72E-06

-2.72E-06

-4.33E-06

-1.77E-07

-3.17E-06

-2.91E-06

-6.94E-06

1.02E-04

b,

-8.57E-12

2.20E-09

282 Chapter 8

0.75

I

u3

•UBOU

"5E

0.25

50

Temperature (°C)

100

Figure 8.8 Effect of temperature on the major food components.


B. Structural ModelsFor heterogeneous foods, the effect of geometry must be considered using

structural models. Utilizing Maxwell's and Eucken's work in the field of electric-ity, Luikov et al. (1968) initially used the idea of an elementary cell, as represen-tative of the model structure of materials, in order to calculate the effective thermalconductivity of powdered systems and solid porous materials. In the same paper, amethod is proposed for the estimation of the effective thermal conductivity of mix-tures of powdered and solid porous materials.

Since then, a number of structural models have been proposed, some ofwhich are given in Table 8.3. The series model assumes that heat conduction isperpendicular to alternate layers of the two phases, while the parallel model as-sumes that the two phases are parallel to heat conduction. In the random model,the two phases are assumed to be randomly mixed. The Maxwell model assumesthat one phase is continuous, while the other phase is dispersed as uniformspheres. Several other models have been reviewed by Rahman (1995), amongothers.

In the mixed model (also called and Krischer model) heat conduction is as-sumed to take place by a combination of parallel and perpendicular heat flow. Thismodel recognizes that there are two extremes in thermal conductivity values, onebeing derived from the parallel model and the other from the series model, whilstthe real value of thermal conductivity should be somewhat in between these twoextremes. A conceptual diagram is shown in Figure 8.9. The distribution factor/isa weighting factor between these extremes. It characterizes the structure of thematerial and it should be independent of material moisture content and tempera-ture,

I-/ Parallel Structure

/ Series Structure

Figure 8.9 The mixed model of thermal conductivity.

284 Chapter 8

Granular (particulate) materials consist of granules (particles) and air, ran-domly packed (Figure 8.10). The induvidual particles consist of solids and water(Figure 8.10). The use of some of these structural models to calculate the thermalconductivity of a hypothetical porous material is presented in Figure 8.11. Theparallel model gives the largest value for the effective thermal conductivity, whilethe series model gives the lowest. All other models predict values in between. Fig-ure 8.12 represents the mixed model for various values of the distribution factor/as a function of the void fraction (porosity).

A systematic general procedure for selecting suitable structural models, evenin multiphase systems, has been proposed by Maroulis et al. (1990). The methodis based on a model discrimination procedure. If a component has unknown ther-mal conductivity, the method estimates the dependence of the temperature on theunknown thermal conductivity, and the suitable structural models simultaneously.

An excellent example of applicability of the above is in the case of starch,an important component of plant foods. The granular starch consists of twophases, the wet granules and the air/vapor mixture in the intergranular space. Thestarch granule also consists of two phases, the dry starch and the water. Conse-quently, the thermal conductivity of the granular starch depends on the thermalconductivities of pure materials (that is, dry pure starch, water, air, and vapor, allfunctions of temperature) and the structures of granular starch and the starch gran-ule. It has been shown that the parallel model is the best model for both the granu-lar starch and the starch granule (Maroulis et al., 1990). These results led to simul-taneous experimental determination of the thermal conductivity of dry pure starchversus temperature. Dry pure starch is a material that cannot be isolated for directmeasurement.


GRANULAR MATERIAL

Particles

GRANULE (PARTICLE)

Figure 8.10 Schematic model of granular materials.

286 Chapter 8

Table 8.3 Structural Models for Thermal Conductivity

Series1 \-e) g

Random

Mixed (Krischer )

=


0.125

0.0250.00

ParallelMaxwell (Continuous phase 1)RandomMaxwell (Continuous phase 2)Series

0.25 0.50

Void Fraction

0.75

Figure 8.11 Structural models for porous materials.

288 Chapter 8

0.125

0.025

Mixed Model (Krischer)Distribution Factor =0.00 (Parallel)0.250.500.751.00 (Series)

0.00 0.25 0.50 0.75

Void Fraction

1.00

Figure 8.12 The mixed (Krischer) model for various values of distribution factor.


V. COMPILATION OF THERMAL CONDUCTIVITY DATA OF FOODS

There is a wide variation of the reported experimental data of thermal con-ductivity of solid food materials, making difficult their utilization in food processand food quality applications. The variation of thermal conductivity in model andreal foods is discussed in Section III of this chapter. The physical structure of solidfoods plays a decisive role not only on the absolute value of thermal conductivity,but also on the effect of moisture content and temperature on this transport prop-erty.

In this section, the thermal conductivity in food materials is approachedfrom a statistical standpoint. Literature data are treated by regression analysis,using the parallel structural model.

Recently published values of thermal conductivity in various foods were re-trieved from the literature, and they were classified and analyzed statistically toreveal the influence of material moisture content and temperature. Structural mod-els, relating thermal conductivity to material moisture content and temperaturewere fitted to all examined data for each material. The data were screened care-fully, using residual analysis techniques. The most promising model was pro-posed, which is based on an Arrhenius-type effect of temperature and it uses aparallel structural model to take into account the effect of material moisture con-tent.

Thermal conductivity data in the literature show a wide variation due to theeffect of the following factors: (a) diverse experimental methods, (b) variation incomposition of the material, (c) variation of the structure of the material. Thermalconductivity depends strongly on moisture, temperature and structure of the mate-rial.

An exhaustive literature search was made in international food engineeringand food science journals in recent years, as follows (Krokida et al., 2001):

• Drying Technology, 1983-1999• Journal of Food Science, 1981-1999• International Journal for Food Science and Technology, 1988-1999• Journal of Food Engineering, 1983-1999• Transactions of the ASAE, 1975-1999• International Journal of Food Properties, 1998-2000

A total number of 146 papers were retrieved from the above journals ac-cording to the distribution presented in Figure 8.13. The accumulation of the pa-pers versus the publishing time is presented in Figure 8.14. The search resulted in1210 data concerning the thermal conductivity in food materials.

290 Chapter 8

J. FoodEngineering

J, of FoodScience

DryingTechnology

Trans of theASAE

Int. J. Food InternationalScience & Journal of

Techn. FoodProperties

Figure 8.13 Number of papers on thermal conductivity data in food materials pub-lished in food engineering and food science journals during recent years.

160

120

oL.

1saZ

o

1970 1980 1990 2000

year

Figure 8.14 Accumulation of published papers on thermal conductivity data forfood materials versus time.


0.001 0.01 0.1 1 10Moisture (kg/kg db)

100

Figure 8.15 Thermal conductivity data for all foods at various moistures.

0.010.1 1 10 100

Temperature (oQ1000

Figure 8.16 Thermal conductivity data for all foods at various temperatures.

292 Chapter 8

These data are plotted versus moisture and temperature in Figures 8.15 and8.16, respectively. These figures show a good picture concerning the range ofvariation of thermal conductivity, moisture and temperature values. More than95% of the data are in the ranges:

• Thermal Conductivity• Moisture• Temperature

0.03 - 2.0 W/mK0.01-65 kg/kg db-43 -160 °C

The histogram in Figure 8.17 shows the distribution of the thermal conduc-tivity values retrieved from the literature. Most of the K values are between 0.1 and1.0 W/mK. Thermal conductivities higher than that of water (0.62 W/mK at 25°C)are characteristic of frozen foods of high moisture content, since the thermal con-ductivity of ice is about 2 W/mK.

The results obtained are presented in detail in Tables 8.4-8.6. More than 100food materials are incorporated in the tables. They are classified into 11 food cate-gories. Table 8.4 shows the related publications for every food material. Table 8.5summarizes the average literature value for each material along with the corre-sponding average values of corresponding moisture and temperature. Table 8.6presents the range of variation of thermal conductivity for each material alongwith the corresponding ranges of moisture and temperature.

•aI'=

oL.

£

1000

100

10

0.01 0.03 0.10 0.30 1.00Thermal Conductivity Values (W/mK)

3.00

Figure 8.17 Histogram of observed values of thermal conductivity in food materi-als.


Table 8.4 Literature for Thermal Conductivity Data in Food Materials:References and Number of Data Retrieved

Material

Baked productsBread

Dough

Soy flour

Cake

Yellow batter

Cup batter

Cereal productsBarley

Corn

Rice

Wheat

Corn meal

Reference

Zanonietal, 1995Zanonietal., 1994Goedeken et a!., 1998

Bouvier et al, 1987Zanonietal, 1995Griffith etal., 1985

Maroulisetal.,1990Wallapapan et al., 1982

Zanonietal., 1995

Baiketal, 1999

Baiketal., 1999

Alagusundaram et al., 1991

Bekeetal, 1994Changetal, 1980Okos etal, 1986

Okos etal, 1986Ramesh, 2000

Changetal., 1980Okos etal., 1986

Laietal, 1992Kumaretal, 1989

Data

6014

536

20389

11742211

1212

7699

21939

1349

1037743

294 Chapter 8

Table 8.4 ContinuedMaterial Reference

Iclli batterMurthyetal, 1997

MaizeHalltdayetal, 1995Tolabaetal.,1988

OatOkosetal.,1986

DairyCheese

Lunaetal., 1985Tavmanetal., 1999

MilkDuaneetal., 1992Duaneetal, 1993Duaneetal, 1994Me Proud eta!., 1983Hori, 1983Zieglereta!., 1985Ready etal, 1993Tavmanetal, 1999Okosetal.,1986

CreamDuaneetal, 1998

ButterTavmanetal, 1999Okosetal.,1986

YogurtKirn etal, 1997Tavmanetal., 1999

WheyOkosetal.,1986

Data

44

119211

13623

12284

11116399

5311523

199

1044



FishCod

Mackerel

Squid

Carp

Surimi

Cake

Shrimp

Calamari

Salmon

FruitsApple

Banana

Reference

Sam et a!., 1987

Sametal., 1987

Rahman et al, 1991Rahman, 1991

Hung el al., 1983

Wangetal., 1990AbuDaggaetal, 1997

Borquezetal, 1999

Karunakar et al, 1998

Rahman, 1991

Sametal., 1987

Ramaswamy elal, 1981Mattea et al., 1989Telis-Romero et al, 1998Rahman, 1991Constenlaetal, 1989Bhumblaetal, 1989Ziegleretal, 1985Mattea etal, 1986Madambaetal, 1995Sheen etal, 1993Buhrietal, 1993Okos etal, 1986Chenetal, 1998

Njieetal, 1998

Data

835555

1612422

3021911

13132299

14382113329

2533211

10911

296 Chapter 8


Peach

Strawberry

Raspberry

Grape

Plantain

Raisin

Pear

Orange

Bilberry

Cherry

LegumesLentils

Reference

Okosetal.,1986

Delgado et al, 1997Bhumbla et al., 1989Okosetal.,1986

Bhumbla etal, 1989Okosetal.,1986

Bhumbla et al, 1989Okosetal.,1986

Njieetal, 1998

Vagenas et al., 1990

Matteaetal, 1989Rahman, 1991Dincer, 1997Mattea et al., 1986Okosetal.,1986

Telis-Romeroetal, 1998Bhumbla etal, 1989Ziegler et al, 1985Okosetal.,1986

Bhumbla et al, 1989Okosetal.,1986

Bhumbla etal, 1989Okosetal.,1986

Alagusundaram et al, 1991

Data

1153112118176644

1532136

159141211211

999



MeatBeef

Chicken

Sausage

Turkey

Mutton

Pork

Pork/soy

Model foodsAmioca

Hylon-7

Reference

Hung etai, 1983Marinos-Kouris et al, 1995Me Proud et al., 1983Perezetal, 1984Rahman, 1991Baghe-Khandan et al, 1982Sanzetal, 1987Califano et al, 1997

Rahman, 1991Sanzetal, 1987

Sheen etal, 1990Ziegleretal, 1987Akterian, 1997

Sanzetal, 1987

Sanzetal, 1987

Sanzetal, 1987

Muzillaetal, 1990

Maroulis etal, 1990Laietal, 1992Drouzasetal, 1991Maroulis et al, 1991Drouzasetal, 1988

Maroulis etal, 1990Laietal, 1992Maroulis et al, 1991Drouzasetal, 1988

Data

1347542292

3025

1927

132

101

12121010111144

28151

718899

439

1996

298 Chapter 8


Potato starch

Starch

Sucrose

Gelatin

Ovalbumin

Tylose

Agar-water

Bentonite-water

Gelatin-water

Amylose

Cellulose gum

Pectin 5%

Pectin 10%

Pectin 5%-glucose 5%

Gelatin-sucrose-water

Glycerin

Reference

Okosetal.,1986

Renaudetal, 1991Njieetal, 1998Maroulis et al, 1991Morley et al, 1997Wangetal.,1993Lanetal, 2000

Renaudetal., 1991Ziegleretai., 1985

Renaudetal, 1991Okosetal.,1986

Renaudetal, 1991Cornillon et al, 1995

Phametal, 1990

Delgado et al, 1997Barringer et al, 1995

Sheen et al, 1993

HalUdayetal.,1995

Voudouris et al, 1995

Saravacos et al,1965

Saravacos et al.,1965

Saravacos el al.,1965

Saravacos et al.,1965

Hallidayetal, 1995

Ryniecki et al, 1993

Data

22

6124

166

186

33303

26242

36241266211111144222222222233



NutsMacadamia

OtherCoconut

Coffee

Soybean

Palm kernel

Lard

Agar-water

Water-NaCI

Water-sucrose

Rapeseed

Tobacco

Sorghum

Sugar

Albumen

NaCl

Reference

Rahman, 1991

Duane et a!., 1995Chenetal, 1998

Sagaraetal., 1994

Okos et al, 1986

Duane etal, 1996

Duane et al., 1997

Wang etal, 1992

Lucas etal, 1999

Lucas etal, 1999

Bilanskietal, 1976Moyseyetal, 1977Okos eta!., 1986

Casadaetal, 1989

Changetal, 1980Okos etal. ,1986

Okos etal, 1986

Okos etal, 1986

Okos etal, 1986

Data

111

13410

19

10101212111

15218102433734

15152233

300 Chapter 8


Honey

Albumine

VegetablesCarrot

Cassava

Garlic

Onion

Pea

Potato

Sugar beet

Turnip

Reference

Okos et al.,1986

Okos eta!., 1986

Niesteruk, 1998Njie et al, 1998Rahman, 1991Buhri et al, 1993

Njieetal, 1998

Madamba et al, 1995

Rapusasetal, 1994

Sastry et al, 1983Alagusundarametal, 1991

Niesteruk, 1996Niesteruk, 1997Niesteruk, 1998Hungetal.,1983Luelal, 1999Njieetal, 1998WangetaL, 1992Rahman, 1991Matteaetal, 1986Hallidayetal, 1995Madamba eta!., 1995Buhri eta!., 1993Cratzeketal.,1993

Niesteruk, 1998Okos etal, 1986

Buhri et al, 1993

Data

121233

15451121663377

1239

45111212

1623921474311



Yam

Beetroot

Parsley

Celery

Tomato

Cucumber

Spinach

Mushrooms

Rutabagas

Radish

Parsnip

Kidney bean

Reference

Njieetal., 1998

Niesteruk, 1998

Niesteruk, 1998

Niesteruk, 1998

Dincer, 1997Choietd.,1983Filkovaetal, 1987Okos eta!., 1986Drouzasetal., 1985

Dincer, 1997

Delgado et al, 1997

Shrivastavaetal, 1999

Buhri et al, 1993

Buhrietal, 1993

Buhri etai, 1993

Zuritz et al, 1989

Data

66221111

311939911

10109911111144

302 Chapter 8

Table 8.5 Thermal Conductivity of Foods Versus Moisture and Temperature:Average Values of Available Data

Material

Baked productsBread-Crust

Crumb

DoughWheat breadRye breadBiscuitSoySoy flourDefatted

Dry defattedCup cake batter

-Yellow cake batter-Cake

"

Cereal productsBarley

SeedsCornDentShelledDustSyrup

Conductivity(W/mK)

0.340.230.270.060.260.340.410.470.400.350.220.440.120.170.170.220.220.250.25

0.290.200.200.390.160.550.090.43

Moisture Temperature(kg/kg db)

0.570.390.350.000.760.890.781.060.070.330.210.360.000.630.630.710.710.560.56

0.670.180.181.550.200.73

0.154.16

(°C)

464447681760232020

150272540151520205151

4000

3636302252

Data

6014923

203323

11

432211

1212

7699

2139

36


Table 8.5 Continued

Material

RicePaddyWheatDustHard red spring

Soft white

Flour

Corn meal-Idli batter-MaizeKernelGritsOat

White

DairyCheeseCheddarMozzarellaCuartirolo Argentina

HamburgerOld Kashkaval

Tulum

Fresh KashkavalBuffet KashkavalFresh cream

Spreadable cheeseLabneLow fat labne

Conductivity(W/mK)

0.150.150.260.070.160.130.59

0.360.360.450.450.27

0.170.290.130.13

0.450.420.350.380.370.390.380.380.400.41

0.430.490.47

0.55


0.220.220.130.150.170.130.100.270.271.711.710.320.160.360.140.14

3.781.200.560.801.200.690.690.690.780.991.291.542.242.94

(°C)

48482622-316

568888161662506527

27

38222323

15232323232323232323

Data

131310322

37744

112911

13623221

222222222

304 Chapter 8

Table 8.5 Continued

Material

Milk-Fresh

Powder

Whole

SkimConcentratedCondensedHalf-halfBaby foodPowdered

Butter-Fat

Yogurt-PlainStrained

Pasterized

Light

Extra lightWhey-CreamPowder

FishCodPerpendicular

MackerelPerpendicular

Conductivity(W/mK)

0.460.460.570.110.460.570.410.490.540.550.300.220.230.21

0.450.560.330.540.580.580.59

0.590.590.130.13

0.791.231.230.800.80

Moisture(kg/kg db)

4.024.009.00

25.333.046.330.924.015.110.030.050.200.180.213.606.252.052.88

4.714.546.589.009.00

44.0044.00

3.294.884.883.423.42

Temperature(°C)

46202325533650544050542123203121402323232340402525

8-10-10

00

Data

8431

318159996

11

523

19292222

4411

835555


Table 8.5 ContinuedConductivity

Material

SquidFreshMantle

DriedTentacle, arrow

TentacleCarp-Surimi-6% cryoprotectant cone12% cryoprotectant cone

Pacific whiting

CakePressed

Shrimppeeled and head removedCalamarimantleSalmonPerpendicular

FruitsApple-RedGreenGolden deliciousGranny Smith

JuiceSauce

(W/mK)

0.350.500.500.240.480.501.211.210.780.850.870.860.590.100.10

1.031.030.510.511.061.06

0.450.450.320.510.410.410.190.530.59


2.365.043.830.873.563.56

0.830.834.024.084.084.083.880.000.00

3.243.244.044.042.402.40

3.733.442.445.600.144.892.173.81

10.11

(°C)

26301530151500

14-2-2-2531515-8-81515

-12-12

3030201545182537

Data

163291122

30777911

13132299

1438225

1143

471

306 Chapter 8

Table 8.5 Continued

Material

BananaDessertPeachFreeze-dried

PlantainFruitsPear-Green

JuiceWilliams

OrangeJuice

BilberryJuiceCherryJuiceGrape-JuiceRaspberryJuiceStrawberryJuice

Tioga

Raisin

-

LegumesLentilsSeeds

Conductivity(W/mK)

0.480.480.040.040.370.370.490.470.520.510.450.410.41

0.550.550.550.550.520.520.550.550.550.630.570.670.230.23

0.220.220.22

Moisture(kg/kg db)

3.123.12

0.980.98

3.222.857.412.602.174.284.288.528.526.526.52

4.083.605.54

7.707.708.44

11.056.701.351.35

0.180.180.18

Temperature(°C)

202035353030342415502527271818

18184250181818141811

4545

222

Data

111166

154263

151522

228622252344

999


Table 8.5 ContinuedConductivity

Material (W/mK)

MeatBeef-FatLeanGroundMinced

Muscle semitendinosusDryfiber

Boneless

Ground round

Whole round

Ground shank

Ground brisket

Whole rib steak

Ground sirloin tipWhole sirloin tipGround ribGround s\viss steakWhole swiss steak

Loaf, uncooked

Loaf heated

ChickenBonelessWhite

0.710.630.540.281.03

1.010.560.310.211.030.510.490.510.440.500.490.480.41

0.510.490.400.471.050.971.33

Moisture Temperature(kg/kg db) (°C)

2.402.222.430.140.63

0.751.960.99

1.033.312.532.322.452.381.842.352.27

1.112.902.822.581.963.463.005.10

1628

1700

11

2025-4606060606060606060601560-9-4

-25

Data

134752322554

20333333333311972

308 Chapter 8

Table 8.5 Continued

Material

Sausage

-ItalianSalami cooked

Lebanon bolognaSalami cottoThuringerSalami Genoa

Salami hard

Pepperoni

TurkeyBonelessMuttonBonelessPorkBonelessPork/soyUnprocessed

Processed

Model foodsAmioca-GelatinizedPowderGranular

Conductivity(W/mK)

0.420.330.930.370.360.370.350.300.320.281.181.180.860.860.930.930.050.050.05

0.630.320.250.510.130.34

Moisture(kg/kg db)

1.021.240.641.701.631.330.960.560.520.372.852.85

2.662.663.253.253.413.173.64

5.263.352.861.23

10.370.12

Temperature(°C)

1822

022222222222222

-12-12-3-3-4-4

252525

305258524840

Data

13421111111

1212

10101111422

28151201696


Table 8.5 Continued

Material

Hylon-7

-GelatinizedPowderGranularPotato starch

GelStarch-GelGelatinized

Hydrated

Granular

Gels

Sucrose

-

GelGelatinGelOvalbumin-

GelXyloseGel

Agar-water

GelGelatin-water-Amylose

Gel

Conductivity(W/mK)

0.330.220.53

0.130.340.040.040.680.101.020.340.380.090.500.850.480.890.960.960.990.881.050.990.990.610.61

0.590.59

0.530.53


5.085.991.90

15.550.120.080.084.710.008.800.100.280.201.653.57

3.723.567.927.927.594.239.283.353.35

36.9536,9565.6765.67

3.503.50

608546484041414225

4504745

100-120-40

0-5-7-4

55

252525253030

Data

4316156622

611

29337

18333

302626361224

66221144

310 Chapter 8

Table 8.5 Continued

Material

Cellulose gumFreeze-dried gel

Pectin 5%Freeze-dried gel

Pectin 10%Freeze-dried gelPectin 5%-glucose 5%Freeze-dried gel

Glycerin

-Gelatin-sucrose-water

"

NutsMacadamiaIntegrifolia

VegetablesCarrot-Large

CassavaRootsGarlic-Onion.

Conductivity(W/mK)

0.060.060.040.040.050.05

0.050.050.470.470.440.44

0.220.220.22

0.430.480.450.520.470.470.360.360.420.42


0.080.080.080.080.080.080.080.083.793.791.441.44

0.020.020.02

3.815.853.828.911.221.220.800.802.052.05

(°C)

41414141414141412020

1515

151515

39222715303015153232

Data

222222223322

1

11

154532663377


Table 8.5 Continued

Material

PeaSeeds

Potato

-MashedFleshGranuleWhiteSpunta

Sugar beet

-RootsTurnip

-YamTubersBeetroot

-Parsley

-Celery

-Tomato-

Juice

PasteCucumber

-Spinach

FreshBlanched

Conductivity(W/mK)

0.220.22

0.450.421.220.54

0.350.530.460.530.560.52

0.480.480.470.470.560.560.17

0.170.150.150.510.610.480.550.620.62

0.380.370.39


0.180.182.352.740.724.540.644.552.173.384.222.750.080.081.451.459.109.102.302.302.302.30

6.2315.60

7.711.73

24.00

24.00

11.0113.66

8.35

22

4957

020621825222520

454530302020202020206821

83402222-2-2-2

Data

99

4525

21

104373411

66221111

311

21911

1055

312 Chapter 8

Table 8.5 Continued

Material

MushroomsPleurotusfloridaRutabagas-Radish-Parsnip

-

Kidney bean

-

OtherCoconutMilkpowder

CoffeeSolutionsSoybeanPowder

WholeCrushedFlourPalm kernelMilkpowder

LardMilkpowderWater-NadSolutionWater-sucroseSolution

Conductivity(W/mK)

0.370.370.450.450.500.50

0.390.39

0.150.15

0.230.150.150.210.21

0.090.080.11

0.100.050.100.100.120.120.460.460.320.32

Moisture(kg/kg db)

3.273.270.080.080.060.060.210.21

0.240.24

2.063.083.081.621.62

0.140.100.130 . 1 10.22

26.0026.0032.0032.00

4.004.000.670.67

Temperature(°C)

555545454545

4545

2020

25373766

34363636262525252510101010

Data

99111111

44

134101010101233

3311111111


Table 8.5 Continued

Material

RapeseedmoleGround

TorchMidas

Crushed

Agar-waterGelTobacco-SugarGlucoseCane sugar

AlbumenFreeze-dried gel

SorghumRs610NC+RS66Grain dust

NaClSolution

Honey-AlbumineSolution

Conductivity(W/mK)

0.110.130.070.100.090.130.620.620.060.060.520.540.510.040.040.240.140.560.090.610.61

0.530.530.410.41


0.100.110.110.100.010.11

19.9019.90

0.260.264.324.804.000.080.080.170.220.160.154.004.00

4.834.830.670.67

141716-41918

303015154244404141

215

36224343

36366060

Data

5221991

1211

33

156922722333

121233

314 Chapter 8

Table 8.6. Thermal Conductivity of Foods Versus Moisture and Temperature:Variation Range of Available DataMaterial

Baked productsBread-Crust

Crumb

DoughWheat bread

Rye bread

Biscuit

SoySoy flourdefatted

dry defatted

Cup cake barter-Yellow cake batter-

Cake

-

Cereal productsBarleySeedsCornDentShelledDustSyrup

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)Min

0.0480.0550.0800.0550.232

0.2300.3270.3960.3900.2300.1060.1800.1060.121

0.1210.2230.2230.0480.048

0.0670.1670.167

0.0850.1420.3710.0850.347

Max

0.6500.5300.5300.0660.2980.6000.5000.600

0.4050.4880.6500.6500.1430.2230.2230.2230.2230.3560.356

0.7400.2250.2250.7400.1750.7400.101

0.513

Min

0.000.000.050.000.720.040.720.850.040.100.000.100.000.550.550.710.71

0.110.11

0.010.110.110.010.010.400.100.23

Max

1.170.820.790.000.82

1.170.821.170.090.600.640.640.000.710.710.710.71

1.221.22

8.090.260.268.090.421.000.208.09

Min

151525151520202020

150202520151520202020

-28-28-28

1036102227

Max

150120100120

18150282020

15060256015

152020

103103

16029297736502277


Table 8.6. ContinuedMaterial

RicePaddyWheatDustHard red springSoft white

Flour

Corn meal-Idli batter-MaizeKernel

Grits

OatWhite

DairyCheeseCheddarMozzarettaCuartirolo Argen-TinoHamburger

Old KashkavalTulumFresh KashkavalBuffet KashkavalFresh creamSpreadable cheeseLabne

Low fat labne


0.0820.0820.0670.0670.1440.1180.4500.2700.2700.3950.3950.0670.1560.0670.1300.130

0.0390.3450.3450.380

0.3720.3810.3680.3770.4030.4060.4330.4760.4630,542

Max

0.3660.3660.6890.0730.1660.1400.6890.4640.464

0.4930.4930.5250.1740.5250.1300.130

0.6860.5480.351

0.383

0.3720.3980.3840.3790.4030.4090.4340.4940.4860.548

Min

0.110.110.010.100.050.01

0.100.180.181.001.000.110.110.160.140.14

0.020.560.560.80

1.200.690.690.690.780.991.291.542.242.94

Max

0.430.430.290.200.290.250.100.430.432.332.330.590.200.590.140.14

44.002.940.560.80

1.200.690.690.690.780.991.291.542.242.94

Min

2020-322-315

432020

151535503527

27

1

151515

15151515151515151515

Max

70706622-31666

160160

20209550952727

90303030

153030303030303030

30

316 Chapter 8


Milk-FreshPowder

mole

SkimConcentrated

Condensed

Half-halfBaby foodPowdered

Butter-Fat

Yogurt

-

PlainStrained

PasterizedLightExtra light

Whey-CreamPowder

FishCodPerpendicular

MackerelPerpendicular


0.1120.3250.5700.1120.2800.4810.3250.3250.4710.4050.1820.0930.2270.0930.0390.5250.0390.5390.5710.5710.5840.547

0.5470.1270.127

0.0400.5490.5490.4090,409

Max

0.6860.5760.5700.1150.6290.6460.4980.6340.6340.6860.5380.3450.2330.3450.6390.6030.6390.540

0.5930.5830.5960.6420.6420.1270.127

1.7201.5431.5431.4281.428

Min

0.021.009.00

22.000.391.50

0.431.00

2.330.030.020.020.180.020.066.250.062.884.714.546.58

9.009.00

44.0044.00

0.004.884.883.423.42

Max

30.009.009.00

30.009.00

19.001.509.009.000.040.140.420.180.426.586.255.662.884.714.546.58

9.009.00

44.0044.00

5.254.884.883.423.42

Min

5202325

55

3523

5355415152011

2515151515

77

2525

-40-22-22-20-20

Max

90202325907565797565543030205540553030303087872525

8033

2020



SquidFresh

Mantle

Dried

Tentacle, arrowTentacleCarp-Surimi-6% cryoprotectant

12% cryoprotectantPacific whitingCakePressedShrimpPeeled and headremovedCalamariMantleSalmonPerpendicular

FruitsApple

-RedGreenGolden deliciousGranny SmithJuiceSauce


0.0400.4900.4830.0400.4750.501

0.7000.7000.4770.4870.4770.4890.524

0.1000.100

0.490

0.4900.5080.5080.4970.497

0.0430.0700.0700.5130.4050.4010.0900.2300.591

Max

0.5070.5000.5070.4400.4750.501

1.7201.7201.5081.4731.5081.4650.7080.1000.100

1.600

1.6000.5170.5171.2451.245

2.2702.2701.5100.5130.4050.4120.2962.2700.591

Min

0.104.753.830.103.563.56

0.830.832.854.084.084.082.850.000.00

1.00

1.004.044.042.032.03

0.140.140.255.600.144.880.500.25

10.11

Max

5.205.203.832.863.563.56

0.830.835.254.084.084.085.250.000.004.20

4.204.044.042.702.70

19.0019.005.995.600.144.894.00

19.0010.11

Min

153015301515

-15-15-40-40-40-40301515

-30

-301515

-24-24

-40-40-4015451525-7

Max

303015301515

151580303030801515

30

30151555

9090451545202590

318 Chapters


BananaDessertPeachFreeze-driedPlantainFruits

Pear-Green

JuiceWilliams

OrangeJuiceBilberryJuiceCherryJuiceGrape-JuiceRaspberryJuice

StrawberryJuiceTiogaRaisin

-

LegumesLentils

Seeds


0.4810.4810.0430.0430.1300.1300.3400.3400.5140.4020.3590.2900.2900.5530.5530.5530.5530.3960.3960.537

0.5440.544

0.5200.5710.5200.1260.126

0.1870.1870.187

Max

0.4810.4810.0430.0430.5200.5200.6290.5570.5330.6290.5050.5600.5600.5540.5540.5540.5540.6390.6390.556

0.5530.5530.9350.571

0.9350.3920.392

0.2530.2530.253

Min

3.123.12

0.160.160.500.507.410.640.500.640.648.528.526.526.520.590.595.54

7.707.706.70

11.056.700.160.16

0.110.110.11

Max

3.123.12

2.002.007.414.907.41

5.674.00

19.0019.008.528.526.526.528.098.095.54

7.707.70

11.0511.056.704.004.00

0.260.260.26

Min

202035353030152315202511

1616

16161620161616

-1516

-154545

-21-21-21

Max

202035353030

8025158025626220202020808020

2020

2820

284545

282828



MeatBeef-FatLeanGround

MincedMuscle semitendi-nosusDry fiber

BonelessGround round

Whole round

Ground shank

Ground brisketWhole rib steakGround sirloin tipWhole sirloin tip

Ground ribGround Swiss steak

Whole swiss steakLoaf, uncookedLoaf, heatedChickenBonelessWhite

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)Min Max Min Max Min Max

0.0490.0950.4540.2640.5100.4000.360

0,0950,1400.4290.4520.4750.4420.4360.4590.4600.4670.3680.4670.4670.4000.4700.4900.4901.268

1.6601.6500.6220.3111.5501.6200.844

0.4900.2431.6500.5900.5040,5980.4580.5520.5180.4940.4500.5750.5080.4000.4701.4521.4521.387

0.010.012.280.100.630.751.11

0.01

0.382.921.991.501.581.36

1.071.611.300.782.161.842.581.962.912.915.10

5.103.692.570.160.630.753.44

2.842.303.692.942.942.923.052.322.922.921.373.443.44

2.581.965.103.225.10

-40-30-18-10-15-15-5

2025

-3030303030

303030303030

1560

-25-20-25

90902015151530

2025309090909090909090909015602020

-25

320 Chapter 8


Sausage-ItalianSalami cookedLebanon bolognaSalami cotto

ThuringerSalami genoa

Salami hardPepperoni

TurkeyBonelessMuttonBoneless

PorkBoneless

Pork/soy

UnprocessedProcessed

Model FoodsAmiocaGelatinizedPowderGranular


0.2750.2830.4700.3700.3550.3650.3450.2950.3150.2750.4900.490

0.3910.3910.4800.4800.0490.049

0.053

0.0380.0800.4320.0800.227

Max

1.3800.3671.3800.3700.3550.3650.3450.2950.3150.2751.6601.660

1.5101.5101.4501.4500.0550.051

0.055

2.3300.6610.661

0.1950.454

Min

0.370.400.641.701.631.330.960.560.520.372.852.85

2.452.453.153.153.083.083.54

0.000.000.010.000.01

Max

1.861.860.64

1.701.631.330.960.560.520.372.852.85

2.802.803.313.313.753.253.75

65.6720.00

3.0020.00

0.23

Min

-1020

-1022222222222222

-24-24

-40-40-30-30252525

-4320202520

Max

22221022222222222222

44

2424

3030252525

1501501357060



Hylon-7Gelatinized

Powder

Granular

Potato starchGelStarch-Gel

GelatinizedHydratedGranular

GelsSucrose-GelGelatinGel

Ovalbumin-GelTyloseGel

Agar-waterGelGelatin-water-

AmyloseGel


0.1000.4420.1000.2270.0390.0390.0610.1000.4800.3300.3640.0610.4360.3500.4050.3500.0390.039

0.4500.4700.450

0.4830.4830.6000.6000.5940.5940.5150,515

0.6610.6610.1600.4540.0410.041

2.1000.1002.1000.3550.3880.1250.5671.7700.5661.7702.0702.0702.3301.7502.330

1.5301.5300.6220.6220.5940.5940.5510.551

0.000.01

11.100.010.020.02

0.000.001.780.100.280.050.660.670.671.000.020.02

2.302.303.20

3.353.35

24.9024.9065.6765.673.003.00

20.004.00

20.000.230.140.14

24.000.00

24.000.100.280.303.009.009.009.00

19.0019.00

19.006.40

19.00

3.353.35

49.0049.0065.6765.67

4.004.00

202025204141

-4225

-4220101580

-4120

-41-41-41

-43-43-42

-30-30202025253030

1507070604141

1202550808075

1203220324141

2620265050303025253030

322 Chapter 8


Cellulose gumFreeze-dried gelPectin 5%Freeze-dried gel

Pectin 10%Freeze-dried gel

Pectin 5%-glucose5%Freeze-dried gelGlycerin

Gelatin-sucrose-water-

NutsMacadamiaIntegrifolia

VegetablesCarrot-Large

CassavaRoots

Garlic-Onion_


0.0560.0560.0380.0380.0440.044

0.04S0.0480.4500.450

0.3960.396

0.2240.2240.224

0.1030.1820.1820.5090.1600.1600.2300.2300.2900.290

Max

0.0630.0630.0390.0390.0470.047

0.0500.0500.4900.490

0.4870.487

0.2240.2240.224

0.6700.6050.6050.5320.5700.5700.4480.4480.5200.520

Min

0.020.020.020.020.020.02

0.020.023.353.35

0.650.65

0.020.020.02

0.060.150.158.910.220.22

0.080.080.320.32

Max

0.140.14

0.140.140.140.14

0.140.144.264.26

2.222.22

0.020.020.02

24.009.009.008.91

2.332.331.651.65

4.154.15

Min

414141414141

41412020

1515

151515

-29151515303015153131

Max

414141414141

41412020

1515

151515

150454515303015153333



PeaSeedsPotato-FleshGranuleWhiteSpuntaSugar beet

-RootsTurnip-YamTubers

Beetroot-

Parsley-Celery

-Tomato-

JuicePasteCucumber-

SpinachFreshBlanched


0.1810.181

0.1200.2090.5360.1200.5190.3310.4480.5350.4480.4800.4800.1600.160

0.5490.5490.1700.1700.1470.1470.2300.6110.2300.4600.6210.6210.347

0.3470.356

0.2560.2560.6430.6430.5360.5790.5360.5500.5890.5850.5890.4800.4800.6000.600

0.5720.5720.1700.1700.1470.1470.6700.6110.6700.6600.6210.6210.4340.4000.434

0.110.110.110.344.540.114.540.501.503.001.500.080.080.190.196.906.902.302.302.302.300.25

15.600.251.16

24.0024.00

8.3513.668.35

4.500.267.337.334.541.44

4.554.005.675.674.000.080.083.763.76

11.3011.302.302.302.302.30

19.8315.6019.832.40

24.0024.0013.6613.66

8.35

-29-21

-15242030152520252045453030

202020202020202120302222

-20-20-20

2828

1301302095202525252045453030

202020202020

15021

150502222212116

324 Chapter 8


MushroomsPleurotusfloridaRutabagas-Radish-Parsnip-Kidney bean

OtherCoconutMilkpowder

CoffeeSolutions

SoybeanPowder

WholeCrushedFlourPalm kernelMilkpowderLardMilkpowder

Water-NaClSolution

Water-sucroseSolution


0.2180.2180.4470.4470.4990.4990.3920.3920.1030.103

0.0390.1150.1150.1530.1530.0400.0660.0950.0850.0400.1020.102

0.1200.1200.4600.4600.3200.320

Max

0.5200.5200.4470.4470.4990.4990.3920.3920.2010.201

0.6560.2170.2170.2770.2770.1330.1040.1330.1260.0610.1020.1020.1200.1200.4600.4600.3200.320

Min

0.110.11

0.080.080.060.060.210.210.120.12

0.010.190.191.221.220.050.100.130 .110.05

26.0026.0032.0032.00

4.004.000.670.67

Max

8.698.690.080.080.060.060.210.210.410.41

32.0026.0026.00

2.512.510.400,100.130 .110.40

26.0026.0032.0032.00

4.004.000.67

0.67

Min

40404545454545452020

-262525

-14-1410101010262525252510101010

Max

707045454545

45452020

905050

2626666666

66262525252510101010



RapeseedWholeGroundTorchMidasCrushed

Agar-waterGelTobacco-

SugarGlucoseCane sugar

AlbumenFreeze-dried gel

SorghumRs6lOGrain dustNaCISolution

Honey-AlbumineSolution


0.0600.1080.0620.0860.0920.0600.6170.6170.0550.0550.3820.4500.3820.0390.039

0.0840.1300.084

0.5680.5680.4400.4400.3820.382

0.1550.1550.0880.1200.0920.0800.6170.6170.0700.0700.6370.6370.6370.0420.042

0.1500.1500.0940.6560.6560.618

0.6180.4250.425

0.010.060.070.010.010.07

19.9019.900.200.200.671.500.670.020.02

0.010.150.104.004.001.501.500.670.67

0.240.150.150.240.010.15

19.9019.900.320.329.008.099.000.140.14

0.300.280.204.004.009.009.000.670.67

-2644

-26

194

30301515020

4141

55

22101022

2727

323232191932

303015158080804141

365

22808071719090

Note: Thermal conductivities higher than that of water (0.62 W/mK at 25°C) are characteristic of frozenfoods of high moisture content, since the thermal conductivity of ice is about 2 W/mK

326 Chapter 8

VI. THERMAL CONDUCTIVITY OF FOODS AS A FUNCTION OFMOISTURE CONTENT AND TEMPERATURE

A concept proposed by Maroulis et al. (2001) is adopted here and applied toobtain an integrated and uniform analysis of the available data. The concept wasapplied simultaneously to all the data of each material, regardless the data sources.Thus, the results are not based on the data of only one author and consequentlythey are of elevated accuracy. A simplified analysis is presented in Chapter 6 forthe moisture diffusivity.

Assume that a material of intermediate moisture content consists of a uni-form mixture of two different materials: (a) a dried material and (b) a wet materialwith infinite moisture. The thermal conductivity is, generally, different for eachmaterial. The thermal conductivity of the mixture could be estimated using a twophase structural model:

1A, Y { 1 / ~rx"

X:(T) (8-13)

where /I (W/mK) the effective thermal conductivity, Ax (W/mK) the thermalconductivity of the dried material (phase a), Axi (W/mK) the thermal conductivityof the wet material (phase b), X (kg/kg db) the material moisture content, and T(°C) the material temperature.

Assume that the thermal conductivities of both phases depend on tempera-ture by an Arrhenius-type model:

= A0 expR(T T

(8-14)

exp (8-15)

where Tr =60°C a reference temperature, R = 0.0083\43kJImolKtheideal gas constant, and A0, /l(., E0, Et are adjustable parameters of the proposedmodel.

The reference temperature of 60°C was chosen as a typical temperature ofair-drying of foods. Thus, the thermal conductivity for every material is character-ized and described by four parameters with physical meaning:


/l0(W/mK) thermal conductivity at moisture X = 0 and temperature T = Tr

At (WlmK} thermal conductivity at moisture X = oo and temperature T = Tr

Ea (kJ I moT) Activation Energy for heat conduction in dry material at X = 0Et (kJ I mol) Activation Energy for heat conduction in wet material at X — co

The resulting model is summarized in Table 8.7 and can be fitted to data using anonlinear regression analysis method.

The model is fitted to all literature data for each material and the estimatesof the model parameters are obtained. Then the residuals are examined and thedata with large residuals are rejected. The procedure is repeated until an acceptedstandard deviation between experimental and calculated values is obtained (Draperand Smith, 1981).

Among the available data only 13 materials have more than 10 data, whichcome from more than 3 publications. The procedure is applied to these data andthe results of parameter estimation are presented in Table 8.8 and in Figure 8.18. Itis clear that thermal conductivity is larger in wet materials.

Figures 8.19-8.36 present retrieved thermal conductivities from the literatureand model-calculated values for selected food materials as a function of moisturecontent and temperature. Thermal conductivity A, tends to increase with the mois-ture content X and the temperature T.

The thermal conductivity parameters /10 and A/, shown in Figure 8.18, varyin the range of 0.05 to 1.0 W/mK. It should be noted that the thermal conductivityof air is about 0.026 W/mK, while that of water is 0.60 W/mK. Values of thermalconductivity of foods higher than 0.60 W/mK are normally found in frozen foodmaterials (Aice=2 W/mK).

The thermal conductivity increases, in general, with increasing moisturecontent. Temperature has a positive effect, which depends strongly on the foodmaterial. The energy of activation for heat conduction E is, in general, higher inthe dry food materials.

328 Chapter 8

Table 8.7 Mathematical Model for Calculating Thermal Conductivity in Foods asa Function of Moisture Content and Temperature

Proposed Mathematical Model

X0expRT T,

X .+——X.expl + X R T T

where /i (W/mK) the thermal conductivity,X (kg/kg db) the material moisture content,T(°C) the material temperature,Tr = 60°C a reference temperature, andR = 0.0083143 kJ/mol K the ideal gas constant.

Adjustable Model Parameters

• Ka(W /mK) thermal conductivity at moisture X = 0 and temperature T = Tr

• "k.(W/ mK) thermal conductivity at moisture X = °o and temperature T = Tr

• E/U / mol) activation energy for heat conduction in dry material at X = 0• E,(kJ/ mol) activation energy for heat conduction in wet material at X = oo


Table 8.8. Parameter Estimates of the Proposed Mathematical Model

No. ofMaterial Papers

Cereal productsCorn

FruitsAppleOrangePear

Model foodsAmiocaStarchHylon

VegetablesPotatoTomato

DairyMilk

MeatBeef

OtherRapeseed

Baked productsDough

3

1245

543

125

5

6

3

3

No. of 4 4, E; E0 sdData (W/mK) (W/mK) (kJ/mol) (kJ/mol) (W/mK)

15

681315

292421

3728

33

37

35

15

1.580

0.5890.6420.658

0.7180.6230.800

0.6110.680

0.665

0.568

0.239

0.800

0.070

0.2870.1060.270

0.1200.2430.180

0.0490.220

0.212

0.280

0.088

0.273

7.2

2.41.32.4

3.20.39.9

0.00.2

1.7

2.2

3.6

2.7

5.0

11.70.01.9

14.40.4

47.05.0

1.9

3.2

0.6

0.0

0.047

0.1140.0070.016

0.0370.0060.072

0.0590.047

0.005

0.017

0.023

0.183

330 Chapter 8

.t! 0,1•Iu•aaoU

0.01

I Moisture=infiniteMoist ure=zero

100

OS

e 10&<u

a.2 i13•u

o.i

• Moisture=infinite0 Moisture=zero

o

Figure 8.18 Parameter estimates of the proposed mathematical model.


Moisture (kg/kg db)

Figure 8.19 Predicted values of thermal conductivity of fruits at 25°C.

332 Chapter 8

Moisture (kg/kg db)

Figure 8.20 Predicted values of thermal conductivity of fruits at 60°C.


Moisture (kg/kg db)

Figure 8.21 Predicted values of thermal conductivity of vegetables at 25°C.

334 Chapter 8

Moisture (kg/kg db)

Figure 8.22 Predicted values of thermal conductivity of vegetables at 60°C.


1.0

Moisture (kg/kg db)

10.0

Figure 8.23 Predicted values of thermal conductivity of miscellaneous foods at25°C.

336 Chapter 8

Moisture (kg/kg db)

Figure 8.24 Predicted values of thermal conductivity of miscellaneous foods at60°C.


Fruits APPLETotal Number of Papers 12


Standard Deviation (sd, W/mK) 0.11Relative Standard Deviation (rsd, %) 142

(93%)

Parameter EstimatesXi (W/mK) 0.59

Xo (W/mK) 0,29Ei (kJ/mol) 2.45

_____Eo (kJ/mol) 11.7

1.0Moisture (kg.kg db)

10.0

Figure 8.25 Thermal conductivity of apple at various temperatures and moisturecontents.

338 Chapter 8

Fruits ORANGETotal Number of Papers 4



(87%)

Parameter EstimatesXi (W/mK) 0.64/u>(W/mK.) 0.11Ei(kJ/mol) 1.26

_____Eo (kJ/mol)______0.0

0.1 1.0Moisture (kg.kg db)

10.0

Figure 8.26 Thermal conductivity of orange at various temperatures and moisturecontents.


Fruits PEARTotal Number of Papers 5

Total Experimental Points 1 5Points Used in Regression Analysis 15 (100%)



Xo (W/mK) 0.27Ei (kJ/mol) 2.45Eo(kJ/mol) 1.9

1 —————————————————————

^EI

"5a•ao

i

0.1 - ————— ———— —— — —— — - - -r —— — - - - - -

i

i

i

AV

££ss

*++

**0

A

\

——— Temperature °C -• 40

^ »60A80

<:&&?u-


E-^••••

, —— L i

>

10.0

Figure 8.27 Thermal conductivity of pear at various temperatures and moisturecontents.

340 Chapter 8

Vegetables POTATOTotal Number of Papers 12


Standard Deviation (sd, W/mK) 0.06Relative Standard Deviation (rsd, %)_____2209

(82%)

Parameter EstimatesW(W/mK) 0.61

Xo (W/mK) 0.05Ei (kJ/mol) 0.00Eo (kJ/mol) 47.0


10.0

Figure 8.28 Thermal conductivity of potato at various temperatures and moisturecontents.


Vegetables TOMATOTotal Number of Papers 5



(90%)


Xo (W/mK) 0.22Ei(kJ/mol) 0.17

____Eo (kJ/mol)______5.0


10.0

Figure 8.29 Thermal conductivity of tomato at various temperatures and moisturecontents.

342 Chapter 8

Model Foods AMIOCATotal Number of Papers 5


Standard Deviation (sd, W/mK) 0.04Relative Standard Deviation (rsd, %)______219

(57%)

Parameter EstimatesXi (W/mK) 0,72Xo(W/mK) 0.12Ei (kJ/mol) 3.22

_____Eo (kJ/mol)______14.4


10.0

Figure 8.30 Thermal conductivity of amioca (starch) at various temperatures andmoisture contents.


Model Foods HYLONTotal Number of Papers 3



(49%)

Parameter EstimatesXi (W/mK) 0.80Xo(W/mK) 0.18Ei (kJ/mol) 9.90

_____Eo (kJ/mol)______0.0

Temperature °C -j• 40


10.0

Figure 8.31 Thermal conductivity of hylon (starch) at various temperatures andmoisture contents.

344 Chapter 8

Model Foods STARCHTotal Number of Papers 4



(44%)


Xo (W/mK) 0.24Ei (kJ/mol) 0.32

____Eo (kJ/mol)_____0.4

1

Temperature °C -• 40 M• 60 IT


10.0

Figure 8.32 Thermal conductivity of starch at various temperatures and moisturecontents.


Dairy MILKTotal Number of Papers 5



(39%)

Parameter EstimatesXi (W/mK) 0.67Xo(W/mK) 0.21Ei(kJ/mol) 1.73

_____Eo (kJ/mol)______1.9


10.0

Figure 8.33 Thermal conductivity of milk at various temperatures and moisturecontents.

346 Chapter 8

Cereal Products CORNTotal Number of Papers 3



(54%)

Parameter EstimatesXi (W/mK) 0.47Xo(W/mK) 0.31Ei (kJ/mol) 0.00

____Eo (kJ/mol)_____9.0


10.0

Figure 8.34 Thermal conductivity of corn (grains) at various temperatures andmoisture contents.


Baked Products DOUGHTotal Number of Papers 3


Standard Deviation (sd, W/mK) 0.18Relative Standard Deviation (rsd, %)_______0

(75%)



____Eo (kJ/mol)______0.0


10.0

Figure 8.35 Thermal conductivity of dough at various temperatures and moisturecontents.

348 Chapter 8

_______________Meat____BEEFTotal Number of Papers 6



(49%)



_____Eo (kJ/mol)______3.2


10.0

Figure 8.36 Thermal conductivity of beef at various temperatures and moisturecontents.


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358 Chapter 8

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Heat and Mass Transfer Coefficientsin Food Systems

I. INTRODUCTION

Heat and mass transfer coefficients are used in the design, optimization, op-eration and control of several food processing operations and equipment. They arerelated to the basic heat and mass transport properties of foods (thermal conductiv-ity and mass diffusivity), and they depend strongly on the food/equipment inter-face and the thermophysical properties of the system. Table 9.1 shows some im-portant heat transfer operations, which are used in food processing. In all of theseoperations, heat must be supplied to or removed from the food material with anexternal heating or cooling medium, through the interface of some type of process-ing equipment. Some operations, such as evaporation, involve mass transfer, butthe controlling transfer mechanism is heat transfer (Heldman and Lund, 1992;Valentas etal., 1997).

Table 9.2 shows some mass transfer operations that are applied to foodprocessing. They are characterized by the removal or separation of a component ofthe food material by the application of heat, e.g. drying, or other driving potential,such as osmosis, reverse osmosis, adsorption, or absorption (King, 1971;Saravacos, 1995). Heat and mass transfer coefficients are empirical transfer con-stants that characterize a given operation from theoretical principles, but they areeither obtained experimentally or correlated in empirical equations applicable toparticular transfer operations and equipment.

Heat transfer coefficients and heat transfer, in general, are used more exten-sively than mass transfer data in most food processing operations. In many cases,mass transfer correlations are similar to correlations developed earlier in heattransfer. In some operations, simultaneous heat and mass transfer may control theprocess, e.g. in the drying of solids.

359

360 Chapter 9

Table 9.1 Heat Transfer Operations in Food ProcessingOperations____________Objective__________________Blanching Enzyme inactivationPasteurization Inactivation of microorganisms and enzymesSterilization Inactivation of microorganismsEvaporation Concentration of liquid foodsRefrigeration Preservation of fresh foodsFreezing Food preservationFrying______________Food preparation______________

Table 9.2 Mass Transfer Operations in Food ProcessingOperations_____________Objective_____________Drying Food preservationExtraction Recovery of componentsDistillation Recovery of volatilesAdsorption Removal/recovery of componentsAbsorption Absorption/removal of gasesReverse osmosis Concentration, desaltingCrystallisation___________Purification of components____

The parallel treatment of heat and mass transfer coefficients is important,since there is an analogy of the two transfer processes, evident in some systems,e.g. air/water, which is based on the transport phenomena.

II. HEAT TRANSFER COEFFICIENTS

A. DefinitionsThe heat transfer coefficient h (W/m2K) at a solid/fluid interface is given by

the equation:

q/A=h(AT) (9-1)

where qlA is the heat flux (W/m2) and /IT is the temperature different (°C or K). Asimilar definition is applicable to liquid/fluid interfaces.

Heat transfer is considered to take place by heat conduction through a filmof thickness L of thermal conductivity /I, according to the equation:

Heat and Mass Transfer Coefficients in Food Systems 361

q/A = QJL)(AT) (9-2)

Thus, the heat transfer coefficient is equivalent to h=UL. However, Eq. (9-2) isdifficult to apply, since the film thickness L cannot be determined accurately be-cause it varies with the conditions of flow at the interface.

The overall heat transfer coefficient U (W/m2K) between two fluids sepa-rated by a conducting wall is given by the equation

q/A = UAT (9-3)

where AT is the overall temperature difference (K).The coefficient U is related to the heat transfer coefficients hi and h2 of the

two sides of the wall and the wall heat conduction x/X by the equation:

\/U=\/h,+x/X+l/h2 (9-4)

where x is the wall thickness (m), and X is the wall thermal conductivity (W/mK).In industrial heat exchangers, the thermal resistance of fouling deposits must beadded in series to the resistances of Eq. (9-4).

The overall heat transfer coefficients are specific for each processingequipment and fluid system, and it is determined usually from experimental meas-urements.

B. Determination of Heat Transfer CoefficientsThe heat transfer coefficient h at a given interface can be determined ex-

perimentally by various methods (Rahman, 1995). In the constant heating (steadystate) method, the heat flux q/A is measured (e.g. by electrical measurement) at agiven temperature difference AT, and the coefficient h is calculated from Eq. (9-1).

In the quasi-steady state method, the heat transfer coefficient is determinedfrom the slope of the heating line of a high conductivity solid, which is assumed toheat uniformly. The heat transfer coefficient can be estimated from the analyticalor numerical solution of the heat conduction (Fourier) equation:

(9-5)dt dX1

where a is the thermal diffusivity of the material.The solution of Eq. (9-5) involves the Biot number for heat transfer, BiH =

(hL/X), from which the heat transfer coefficient can be estimated. The heat transfercoefficient h at the interface of processing equipment can be measured by the heatflux sensors method, which simultaneously measures the surface temperature andthe heat flux (Karwe and Godavarti, 1997). The sensors consist of a differential

362 Chapter 9

thermopile of thermal resistance with two inserted thermocouples. They aremounted on the heat transfer surface by a high thermal conductivity paste.

Approximate values of h can be estimated indirectly by measuring the pa-rameters of some physical processes, which involve heat transfer, such as thefreezing time of a material (Plank's equation) or the evaporation rate of a liquid ina flat surface at a given temperature difference A T.

Special experimental arrangements are required for the estimation of theheat transfer coefficients between particles and a liquid, both in motion, as in asep-tic processing of food suspensions. The particle temperature may be measured bya moving thermocouple or estimated from the change of color of special materials,such as liquid crystals coated on acrylic spherical particles and observed through atransparent flow tube.

C. General Correlations of the Heat Transfer CoefficientCorrelations of heat transfer data are useful for estimating the heat transfer

coefficient h in various processing equipment and operating conditions. Thesecorrelations contain, in general, dimensionless numbers, characteristic of the heattransfer mechanism, the flow conditions, and the thermophysical and transportproperties of the fluids. Table 9.3 lists the most important dimensionless numbersused in both heat and mass transfer operations.

The Reynolds number (Re=uL/v) is used widely in almost all correlations.In this number, the velocity u is in (m/s), the length I is in (m) and the kinematicviscosity or momentum diffusivity (v=rj/p~) is in (m2/s). The length L can be theinternal diameter of the tube, the equivalent diameter of the noncircular duct, thediameter of a spherical particle or droplet, or the thickness of a falling film. Somedimensionless numbers, used in both heat and mass transfer correlations, are de-noted by the subscripts H and M respectively, i.e. Bin Bi^, Stn, St^, JH andy^/-

Table 9.4 shows some heat transfer correlations of general applications. Fornatural convection, the parameters a and m characterize the various shapes of theequipment and the conditions of the fluid (McAdams, 1954; Perry and Green,1984; Geankoplis, 1993; Rahman, 1995).

The ratio of tube diameter to tube length D/L is important in the laminarflow (Re < 2100), but it becomes negligible in the tubular flow in long tubes (L/D> 60). For shorter tubes, the ratio D/L should be included in the correlation.

The viscosity ratio r\/r]w refers to the different viscosity in the bulk of thefluid 77 and at the tube wall t]w. This ratio becomes important in highly viscousfluids, like oils, in which the viscosity drops sharply at the high wall temperatures,increasing the heat transfer coefficients. Several other correlations have been pro-posed in the literature for different heat transfer of fluid systems, like flow outsidetubes and flow in packed beds.


The heat transfer coefficients of condensing vapors have been correlated tothe geometry of the tubes and the properties of the liquid film or droplets. Veryhigh heat transfer coefficients are obtained by drop-wise condensation.

Table 9.3 Dimensionless Numbers in Heat and Mass Transfer CalculationsNumberReynoldsNusseltPrandtlGrashofGraetzBiotSherwoodSchmidtStantonStantonLewisPecletBiotHeat Transfer FactorMass Transfer Factor

Re=uL/vNu=hL/APr=v/aGr=L3g(Ap/p)/v2

Gz=GACpMBiH=hmSh=kcL/DSc=v/DStH=h/G CpStM=kc/uLe=a/DPe=uL/DBiM=kcL/XJH=StHPr2'^jM=StMSc /s

ApplicationsFlow processesHeat transferHeat transferFree convectionHeat transferHeat transferMass transferDiffusion processesHeat transferMass transferHeat/mass transferFlow/diffusionMass transferHeat transferMass transfer

A, interfacial area (m2); L, length (m); a thermal diffusivity (m2/s); Cp, specific heat (J/kg K); D massdiffusivity (m2/s); g acceleration of gravity (m2/s); G=up, mass flow rate kg/m2s; h, heat transfer coeffi-cient (W/m2K); kc, mass transfer coefficient (m/s); 77 viscosity (Pas); p, density (kg/m3); u, velocity(m/s)

Table 9.4 General Heat Transfer Correlations____________________________Transfer System________________Correlation_____________Natural convection Nu = a (Gr Pr)m

Laminar inside tubes Nu = l.B6[RePr(D/L)f\rj/rjwfu

Turbulent inside long tubes Nu = O.Q23Re°*Pr1'3 (rj/tjw}°M

Parallel to flat plate (laminar) Nu = 0.664/?e°5 Pr113

Parallel to flat plate (turbulent) Nu = 0.0366#e°8 Pr1'3Past single sphere_______________Nu = 2.0 +0.60Re°'5 Pr113_______Dimensionless numbers defined in Table 9.3. a and m, parameters of natural convection characteristicof the system (Perry and Green, 1984); L, D length and diameter of tube. Long tubes L/D>60

364 Chapter 9

D. Simplified Equations for Air and Water

The heat transfer coefficients of air and water in some important operationscan be estimated from simplified dimensional equations, applicable to specificequipment geometries and system conditions (Perry and Green, 1984; Geankoplis,1993):

a. Natural convection of air:Horizontal tubes, h = 1 .42 (A T/d0) 1/4 (9-6)Vertical tubes, h=\A2 (AT/L)W (9-7)b. Air in drying (constant rate):Parallel flow, h = 0.0204G0'8 (9-8)Perpendicular flow, h=l.ll G°'37 (9-9)

c. Falling films of water:/j = 9150r1/3 (9-10)

d. Condensing water vapors:Horizontal tubes, h = 10800 / [(Nd0f\AT)l/3] (9-11)Vertical tubes, h = 13900 / [Lw(AT)m] (9-12)

where AT is the temperature difference (K), d0 is the outside diameter (m), L is thelength (m), G is the mass flow rate (kg/m2s), F is the irrigation flow rate of thefilms (kg/m s) and N is the number of horizontal tubes in a vertical plane.

III. MASS TRANSFER COEFFICIENTS

A. DefinitionsMass transfer in industrial and other applications is usually expressed by

phenomenological mass transfer equations, instead of the basic mass diffusionmodel. The mass transfer equations use lumped parameters and average concentra-tion, while the diffusion model has distributed parameters for the dependent vari-able (concentration), which can vary with the independent variables of distanceand time (Cussler, 1997).

The mass transfer coefficients are functions of the mass diffusivity, the vis-cosity, the velocity of the fluid, and the geometry of the transfer systems. Themass diffusivity, in the diffusion model, is a fundamental property based on mo-lecular interactions and on the physical structure of the material.

The mass transfer coefficient kc (m/s) in a process is defined in an analo-gous manner with the heat transfer coefficient:

J = kcAC (9-13)


where J is the mass flux (kg/m2s) and AC is the concentration difference (kg/m3).In contrast to heat transfer where the driving force is the temperature differ-

ence AT, m mass transfer the driving force can be expressed by the concentrationdifference AC, the difference of mass fraction AY, or the pressure difference AP.Thus, three mean mass transfer coefficients can be defined by the following equa-tion (Saravacos, 1997):

J = kcAC = kYAY = kpAP (9-14)

The units of the three mass transfer coefficients depend on the units of AC, AY andAP and they are usually kc (m/s), kY (kg/m2s) and kp (kg/m2sPa). In food engineer-ing and especially in drying calculations, the symbol hM is used instead of kY, withthe same units (kg/m2s).

In an analogy with the overall heat transfer coefficient K, the overall masstransfer coefficient is used to express mass transfer through the interface of twofluids, according to the equation:

l/K=\fkci+\/kc2 (9-15)

where kcl and kC2 are the mass transfer coefficients of the two contacting fluids. Itshould be noted that in mass transfer there is no wall resistance and the two fluidsat the interface are assumed to be in thermodynamic equilibrium.

Volumetric mass transfer coefficients (kcv) may be used in some industrialoperations, defined by the equation:

kcv=a.kc (9-16)

where a = A/Vis the specific surface of the transfer system (m2/m3). Thus the unitsof kcv will be (1/s) and of h

B. Determination of Mass Transfer Coefficients

The mass transfer coefficients can be determined by direct or indirect meas-urement of the mass transfer rates in a controlled experimental system.

The wetted wall column has been used to determine ^-values in liquid/gasand liquid/vapor systems, like absorption of gas in aqueous solutions (Sherwoodet al., 1975; Brodkey and Hershey, 1988). The mass flux is measured at a givendriving force (AC, AY or AP) and the corresponding coefficients (kc, kY or kp) aredetermined.

The mass transfer coefficients (kc or hM) during the constant rate period ofdrying can be estimated from the drying rate of a known sample at well-defined

366 Chapter 9

drying conditions. As an illustration, the mass transfer coefficient in the air dryingof spherical starch samples 21 mm diameter at 60°C, 10% RHand 2 m/s air veloc-ity was determined as kc= 34 mm/s (Saravacos et al, 1988).

It should be noted that the drying rate of wet high moisture samples is closeto the evaporation of water from a free surface. However, in drying food materials,some resistance to mass transfer is usually present at the interface and in the inte-rior of the product, resulting in significantly lower drying rates. Thus, the masstransfer coefficient in drying grapes is lower, e.g. 7 mm/s or 13 mm/s, dependingon skin resistance to moisture transfer.

The mass transfer coefficient during drying kY or hM can be estimated simul-taneously with the heat transfer coefficient h and the moisture diffusivity D fromdrying data (Marinos-Kouris and Maroulis, 1995). The experimental drying dataare fitted by regression analysis to a heat and mass transfer model, assuming cer-tain empirical relationships. The results, obtained for the heat and mass transfercoefficients, are much lower than the values of evaporation of water from freesurfaces, since during drying the heat and mass transfer interface moves inside theporous solid food material, becoming much larger than the outside surface of thematerial.

C. Empirical CorrelationsTables 9.5 and 9.6 show some empirical correlations of the mass transfer

coefficient (kc) in fluid/solid and fluid/fluid systems. Fluid/solid systems are com-mon in drying of solids, solvent extraction of solids and adsorption operations.Fluid/liquid interfaces are important in aeration, de-aeration, and carbona-tion/decarbonation of liquid foods.

Table 9.5 Mass Transfer Correlations for Fluid/Solid InterfacesTransfer system_________________Correlation_______Membrane Sh = 1Laminar inside tubes Sh = 1.62 (cfuILD)1/3

Turbulent inside tubes Sh = 0.026 Re°8Sc1'3Parallel to flat plate (laminar) Sh = 0.646 Re0'5 Sc>/3

Past single sphere Sh = 2.0 + 0.60 Re°'5Scl/3

Packed beds Sh=\.ll Re°A2 (1 /Sc)2/3

Spinning disc__________________Sh = Q.62Re°'5Sc1'3Dimensionless numbers defined in Table 9.3.


Table 9.6 Mass Transfer Correlations for Fluid/Fluid InterfacesTransfer system____________Correlation________________Gas bubbles in unstirred tank Sh = OA2 Gr1/45c"3

Gas bubbles in stirred tank Sh=l.62 [(P/V) cflpP3]1/4 5c1/3

Small liquid drops in unstirred solution Sh = 1.13 (dulD)°'%Falling films______________Sh = 0.69 (zu/Df5_____________Dimensionless numbers defined in Table 9.3; d, drop diameter (m); z, position along film (m); P/Vstirrer power per volume.

D. Theories of Mass TransferThe empirical mass transfer data, used in various correlations can be inter-

preted in terms of approximate or exact theories of mass transfer. The mass trans-fer theories were developed mainly for fluid/fluid systems. The most importanttheories are briefly the following (Cussler, 1997).

1. Film TheoryThe mass transfer coefficient kc is a function of the first power of the diffu-

sion coefficient £>:

hc =D/L (9-17)

where L (m) is the film thickness, which is difficult to determine accurately, sinceit is a function of the flow conditions, the geometry of the system, and the physicalproperties of the fluid.

2. Penetration TheoryThe mass transfer coefficient kc is a function of the square root of the mass

diffusivity D:

kc=2(Du/nLf2 (9-18)

where L is the depth of penetration (m) and u is the velocity (m/s) of penetration.The contact time between the diffusivity components and the fluid is defined asu/L, and it is difficult to determine experimentally.

3. Surface Renewal TheoryThe mass transfer coefficient kc is a function of the square root of mass dif-

fusivity D, in a similar manner with the penetration theory:

368 Chapter 9

kc=(Drf2 (9-19)

where T is the average time for a fluid element in the interface region.

4. Boundary Layer TheoryThe boundary layer theory, applied primarily in fluid mechanism and heat

transfer, gives a more accurate correlation of the mass transfer coefficient kc in thelaminar flow. The kc is a function of the 2/3 power of mass diffusivity D.

The average mass transfer coefficient kc, past a flat plate of length L, isgiven by the following empirical equation, which is analogous to the correspond-ing heat transfer relationship:

kc = 0.00646 (D/L)RelK So213 (9-20)

where the Reynolds number is defined as Re=Lu/v.The heat and mass transfer analogies are useful in evaluating the heat/mass

transfer mechanisms and in estimating and inter converting the heat and masstransfer coefficients. The Chilton Colburn (or Colburn) analogy for heat and masstransfer indicates that in fluid systems, under certain conditions, the heat and masstransfer factors are equal (Geankoplis, 1993; Saravacos, 1997):

JH=JM (9-21)

where jH= 5^/'r2/3,yw= StMSc2n and StH= h/upCp, StM= kc/u or StM= h^upThe Colburn analogy in air/water mixtures (applications in drying and air

conditioning) is simplified, since the Pr and Sc are approximately equal (Pr = Sc= 0.8). Therefore, we may have StH = StM or h/upCp = kC/u or h/pCp =kc,

In terms of the mass transfer coefficient hM, the last relationship becomes:

h/Cp = hM (9-22)

The specific heat of atmospheric air at ambient conditions is approximatelyCp = 1000 J/kgK. Therefore, Eq. (9-22) yields h = 1000hM, where h is in W/m2Kand hM in kg/m2s. If the units of hM are taken as g/m2s, the last relationship is writ-ten as (Saravacos, 1997):

Atmospheric air, h (W/m2K) = HM (g/m2s) (9-23)

A similar relationship is obtained between the coefficients h and kc'.

Atmospheric air, h (W/m2K) = kc (mm/s) (9-24)


IV. HEAT TRANSFER COEFFICIENTS IN FOOD SYSTEMS

The heat and mass transfer coefficients in food systems are determined ex-perimentally or correlated empirically from pilot plant and industrial data. Theyare specific for each food process and processing equipment and are related to thephysical structure of the food materials.

Most of the literature data refer to heat transfer coefficients, since heat trans-fer is the rate controlling mechanism in many processing operations. Mass transfercoefficients can be related to heat transfer in some important operations, like dry-ing, using the Colburn analogy of heat and mass transfer.

Typical values of heat transfer coefficients are shown in Table 9.7 (Hall-strom et al., 1988; Perry and Green, 1997; Rahman, 1995; Saravacos, 1995). De-tailed data and empirical correlations for both transfer coefficients are presented insections VI and VII of this chapter.

A. Heat Transfer in Fluid FoodsHeat transfer in viscous non-Newtonian fluids in laminar flow in tubes is

expressed by a correlation analogous to the equation for Newtonian fluids:

(9-25)

where the Graetz number Gz = GrCp/AL, and G is the mass flow rate (kg/m2s).

Table 9.7 Typical Heat Transfer Coefficient h and Overall Coefficients U inFood Processing Operations_____________________ ______

Heat Transfer System h, W/m2KAir/process equipment, natural convention 5 - 20a

Baking ovens 20 - 80a

Air drying, constant rate period 30 - 200a

Air drying, falling rate period 20 - 60Water, turbulent flow 1000 - 3000Boiling water 5000 - 10000Condensing water vapor 5000 - 50000Refrigeration, air cooling 20 - 200Canned foods, retorts 150 - 500Aseptic processing, particles 500 - 3000Freezing, air/refrigerants 20-500Frying, oil/solids 250 - 1000Heat exchangers (tubular/plate) 500 - 3500 (overall U)Evaporators____________________500 - 3000 (overall U)

' Similar numerical values for the mass transfer coefficients kc (mm/s) or hM (g/m2s), applying theColbum analogy.

370 Chapter 9

The apparent viscosities at the bulk of the fluid and at the wall tja and aredetermined for the given shear rate y using the Theological constants K and n of thefluid for a mean temperature. Heat transfer in agitated vessels is expressed by theempirical correlation (Saravacos and Moyer, 1967):

= CRe°'66Pr1/3Ola/Tlaw)'

,0.14 (9-26)

where the coefficient C = 0.55 for Newtonian and C = 1.474 for non-Newtonianfluids.

The Reynolds number is estimated as Re = (d2Np)lrja where d is the diameterof the impeller, and rja is the apparent viscosity estimated at the agitation speed TVas r\a = Ky"'1 where K and n are the Theological constants of the fluid at the meantemperature. The shear rate y for the pilot-scale agitated kettle, described in thisreference (0.40 m diameter, anchor agitator), was calculated from the empiricalrelation 7= 13N.

The heat transfer coefficients h at the internal interface of the vessel for asugar solution and for applesauce increased linearly with speed of agitation(RPM), as shown in Figure 9.1.

Figure 9.2 shows that the overall heat transfer coefficient U in the agitatedkettle decreases almost linearly when the flow consistency coefficient K is in-creased.

10000

1000 --

Figure 9.1 Heat transfer coefficients in agitated kettle. S, sucrose solution 40° Brix;A, applesauce; RPM, 1/min


1600

1300

1000

K (Pa s")

Figure 9.2. Overall heat transfer coefficient (U) of fruit purees in agitated kettle.K, flow consistency coefficient.

10

B. Heat Transfer in Canned FoodsSeveral heat transfer correlations for canned foods are presented by Rahman

(1995). In most cases of heating/cooling of cans, the product heat transfer coeffi-cient ht is controlling the transfer process, since the outside (heating/cooling me-dium) coefficient and the heat conductance of the wall l/x are generally high (me-tallic or glass containers). However, heat transfer in plastic containers may be con-trolled by the wall thermal resistance, due to the low thermal conductivity and thehigh wall thickness of the plastic material Eq. (9-4).

The Reynolds numbers for Newtonian fluids is estimated as Re =where d is the can diameter and N is the speed of rotation (1/s) of the can. For non-Newtonian fluids, the dimensionless numbers used are the following (Rao, 1999):

Re = - 4nl3n + l

(9-27)

372 Chapter 9

4n

Gr 2 2 - 2 V ;

where K and n are the rheological constants of the fluid at a mean temperature, andfi = (A V/AT)IV, 1/K (natural convection).

Heat transfer in cans in an agitated retort (Steritort) is considered as the sumof the contributions of both natural and forced convection:

Nu = A[(Gr}(Prf + C^Re\Pr\D / L)]D (9-30)

where, for Newtonian fluids, A = 0.135, B = 0.323, C = 3.91xlO"3, and D = 1.369and for non-Newtonian fluids, A = 2.319, B = 0.218, C = 4.1xlO'7, and D = 1.836

In end-over-end agitated cans the following correlations were obtained(Rao, 1999):

Nu = 2 .9 Re°A36 Pr°2*7 for Newtonian fluids (9-3 1 )

Nu = Re°ABS Pr°'361 for non-Newtonian fluids (9-32)

Non-Newtonian biopolymers, when subjected to extreme heat treatment, suffersignificant losses in apparent viscosity.

C. Evaporation of Fluid FoodsHeat transfer controls the evaporation rate of fluid foods and high heat trans-

fer coefficients are essential in the various types of equipment. Prediction of theheat transfer coefficients in evaporators is difficult, and experimental values of theoverall heat transfer coefficient U are used in practical applications.

The overall heat transfer coefficient is a function of the two surface heattransfer coefficients //,• and h0, the wall thermal conductance MX, and the foulingresistance Eq. (9-33):

-1 = 1 + - + — + FR (9-33)U h k h,

The fouling resistance 7-7? becomes important in the evaporation of liquid foodscontaining colloids and suspensions, which tend to deposit on the evaporatorwalls, reducing significantly the heat transfer rate.


10000

Figure 9.3 Overall heat transfer coefficients U in evaporation of clarified CL and unfil-tered UFT apple juice at 55°C.

Falling film evaporators are used extensively in the concentration of fruitjuices and other liquid foods because they are simple in construction and they havehigh heat transfer coefficients. Figure 9.3 shows overall heat transfer coefficientsU for apple juices in a pilot plant falling film evaporator, 5 cm diameter and 3 mlong tube (Saravacos and Moyer, 1970).

Higher U values were obtained in the evaporation of depectinized (clarified)apple juice (1200 to 2000 W/m2K) than the unfiltered (cloudy) juice, which tendedto foul the heat transfer surface as the concentration was increased. The U valuefor water, under the same conditions was higher as expected: U= 2300 W/m2K.

D. Improvement of Heat / Mass TransferJet impingement ovens and freezers operate at high heat transfer rates, due

to the high air velocities at the air/food interface. Heat transfer coefficients of 250-350 W/m2K can be obtained in ovens, baking cookies, crackers and cereals (NitinandKarwe, 1999).

Ultrasounds can substantially improve the air-drying rate of porous foods,like apples (acoustically-assisted drying). Ultrasound of 155-163 db increased themoisture diffusivity at 60°C from 7xlO'10 to 14xlO'10 m2/s (Mulet et al., 1999).

374 Chapter 9

V. HEAT TRANSFER COEFFICIENTS IN FOOD PROCESSING: COM-PILATION OF LITERATURE DATA

Recently reported heat transfer coefficient data in food processing were re-trieved from the following journals (Krokida et al., 200 la):

• Drying Technology, 1983-1999• Journal of Food Science, 1981-1999• International Journal for Food Science and Technology, 1988-1999• Journal of Food Engineering, 1983-1999• Transactions of the ASAE, 1975-1999• International Journal of Food Properties, 1998-2000

A total number of 54 papers were retrieved from the above journals. Thedata refer to 7 different processes (Table 9.8) and include about 40 food materials(Table 9.9). Most of the data were available in the form of empirical equationsusing dimensionless numbers. All available empirical equations were transformedin the form of heat transfer factor versus Reynolds number (jH = aRe"). This equa-tion was also fitted to all data for each process and the resulting equations charac-terize the process, since they are based on the data from all available materials.

The results are classified by process and material and are presented in Table9.10. All the equations are presented in Figure 9.4 to define the range of variationof they'// and Re. The range of variation by process is also sketched in Figure 9.5.The above results are presented analytically for each process in Figures 9.6-9.11.The effect of food material is obvious in these diagrams. The results of fitting theequation to all data for each process is summarized in Table 9.11 and in Figure9.12.

Heat transfer coefficient values for process design can be obtained easilyfrom the proposed equations and graphs. The range of variation of this uncertaincoefficient can also be obtained in order to carry out valuable process sensitivityanalysis. Estimations for materials not included in the data can also be made usingsimilar materials or average values. It is expected that the resulting equations aremore representative and predict more accurately the heat transfer coefficients.


Table 9.8 Number of Available Equations for each Food Process

1

2

3

4

5

6

7

ProcessBakingForced convectionBlanchingSteamCoolingForced convectionDryingConvectiveFluidized bedRotaryFreezingForced convectionStorageForced convectionSterilizationAsepticRetortTotal No. of equations

No. of equations

1

1

9

1614

6

4

93

54

376 Chapter 9

Table 9.9 Number of Available Equations for each Food Material

123456189

101112131415161718192021222324252627282930313233343536

MaterialApplesApricotsBarleyBeefCakesCalcium alginate gelCanola seedsCarrotCornCorn starchFigsFishGrapesGreen beansHamburgerMaizeMaltMeat carcassModel foodNewtonian liquidsNon-food materialParticulate liquid foodsPeachesPotatoesRaspberriesRiceSoyaSoybeanStrawberriesSugarWheatSpherical particlesTomatoesCorn creamRapeseedMeatballsTotal No. of equations

No. of Equations112111112111312111413312112111311111

54


Table 9.10 Parameters of the Equation jH = aRe" for each Process and each Mate-rial

Process/product/reference a „ mm Re max Re

BakingCakes

Baiketal., 1999

BlanchingGreen Beans

Zhangetal., 1991

CoolingApples

Fikiinetal., 1999Apricots

Fikiinetal., 1999Figs

Dincer, 1995Grapes

Fikiinetal., 1999Model Food

Alvarez et al., 1999Peaches

Fikiinetal., 1999Raspberries

Fikiinetal., 1999Strawberries

Fikiinetal., 1999Tomatoes

Dincer, 1997

0.801

0.00850

0.0304

0.114

8.39

0.472

2.93

0.186

0.0293

0.136

0.267

-0.390

-0.443

-0.286

-0.440

-0.492

-0.516

-0.569

-0.500

-0.320

-0.440

-0.550

40

150

4,000

2,000

3,500

1,300

2,000

3,700

1,300

1,900

1,000

3,000

1,500

48,000

25,000

9,000

17,000

12,000

43,000

16,000

25,000

24,000

378 Chapter 9

Table 9.10 ContinuedProcess/product/reference a n minRe ma\Re

Drying

ConvectiveBarley

Sokhansanj, 1987Canola Seeds

Langetal., 1996Carrot

Mulct etal., 1989Corn

Fortes etal., 1981Torrezetal., 1998

GravesGhiausetal, 1997

Vagenas et al, 1990Maize

Mourad et al., 1997Malt

Lopezetal., 1997Potatoes

Wangetal., 1995Rice

Torrezetal., 1998Soybean

Taranto et al., 1997Wheat

Langetal., 1996Sokhansanj, 1987

Fluidized bedCorn Starch

3.26

0.458

0.692

1.064.12

0.6650.741

11.9

0.196

0.224

4.12

2.48

1493.26

-0.650

-0.241

-0.486

-0.566-0.650

-0.500-0.430

-0.901

-0.185

-0.200

-0.650

-0.523

-0.340-0.650

20

30

500

40020

81,000

150

60

2,000

20

200

5020

1,000

50

5,000

1,1001,000

503,000

1,500

80

11,000

1,000

1,500

1001,000

Shu-De etal., 1993 0.101 -0.355 3,200 13,000


Table 9.10 ContinuedProcess/product/reference n min Re max Re

RotaryFish

Sheneetal, 1996 0.00160Soya

Alvarez et al., 1994 0.00960Sheneetal., 1996 0.000300

SusarWangetal., 1993 0.805

Freezing

-0.258

-0.587-0.258

80

1020

70

300

10080

-0.528 1,500 17,000

BeetHeldman, 1980

Calcium alsinate selSheng, 1994Hamburser

Floresetal, 1988Toccietal., 1995

Meat carcassMallikarjunan et al., 1994

MeatballsToccietal., 1995

Storage

0.650

48.6

8.874.67

0.228

0.536

-0.418

-0.535

-0.672-0.645

-0.269

-0.485

80

300

7,5009,000

1,800

3,400

25,000

600

150,00073,000

20,000

28,000

Potatoes90Xuetal., 1999 0.658 -0.425

WheatChangetal, 1993 0.0136 -0.196 1,500 10,000

380 Chapter 9

Table 9.10 ContinuedProcess/product/reference // min Re max Re

Sterilization

AsepticModel food

Balasubramaniam et al, 1994Sastryetal., 1990Zuritzetal, 1990

Non-food materialKramers, 1946

Ranzetal.,1952Whitaker, 1972

Paniculate liquid foodsMankadetal., 1997

Sannervik et al., 1996Spherical particlesAstrometal., 1994

0.5000.448

3.42

0.7480.6620.517

0.2250.0493

2.26

-0.507-0.519-0.687

-0.512-0.508-0.441

-0.400-0.199

-0.474

5,0002,4002,000

3,0003,0003,000

1401,800

4,300

20,00045,00011,000

85,00085,00085,000

1,5005,200

13,000

RetortNewtonian liquids

Anantheswaran et al., 1985Particulate liquid foods

Sablanietal, 1997Corn cream

Zamanetal., 1991

2.74 -0.562 11,000 400,000

0.564 -0.403 30 1,600

0.108 -0.343 130,000 1,100,000


Table 9.11 Parameters of the Equation jH = a Re" for each Process

Process

Baking

Blanching

Cooling

Drying /convective

Drying /fluidized bed

Drying /rotary

Freezing

Storage

Sterilization /aseptic

Sterilization /retort

a

0.80

0.0085

0.143

1.04

0.10

0.001

1.00

0.259

0.357

1.034

«

-0.390

-0.443

-0.455

-0.455

-0.354

-0.161

-0.486

-0.387

-0.450

-0.499

mm Re

40

150

1,000

8

3,200

10

80

70

140

30

max Re

3,000

1,500

48,000

11,000

13,000

300

150,000

10,000

45,000

110,000

The data of Tables 9.10 and 9.11 demonstrate the importance of the flowconditions (Reynolds number, Re) and the type of food process and product on theheat transfer characteristics (heat transfer factor, jH). As expected from theoreticalconsiderations and experience in other fields, the heat transfer factor, jH decreaseswith a negative exponent of about -0.5 of the Re. The highest jH values are ob-tained in drying and baking operations, while the lowest values are in storage andblanching. Granular food materials, such as corn and wheat appear to have betterheat transfer characteristics than large fruits (apples).

Regression analysis of published mass transfer data show the similarity be-tween the heat transfer factory'// and the mass transfer factor jM (see section VI ofthis chapter).

382 Chapter 9

JH

0.01

0.001

0.0001

0.000011 10 100 1000 10000 100000 1000000 10000000

Re

Figure 9.4 Heat transfer factor jH versus Reynolds number Re for all the examined proc-esses and materials.


JH 0.01

0.001

0.000110 1 000 10 000

Re100 000 1 000 000

Figure 9.5 Ranges of variation of the heat transfer factor^ versus Reynolds number Refor all the examined processes.

384 Chapter 9

0.0011 000 10000 Re 100000

Figure 9.6 Heat transfer factory'// versus Reynolds number Re for cooling process andvarious materials.


JH 0.1

0.0110 100 1000 10000

Re

Figure 9.7 Heat transfer factor jH versus Reynolds number Re for convective dryingprocess and various materials.

386 Chapter 9

JH0.l

0.01

0.001100 000

Figure 9.8 Heat transfer factor jH versus Reynolds number Re for freezing process andvarious materials.


0.1

JH

0.01

0.00110

Storage i

WlhesiT

100 1000 10000 100000Re

Figure 9.9 Heat transfer factory'// versus Reynolds number Re for storage process andvarious materials.

388 Chapter 9

0.1

JH

0.01

0.001

Non-Foa4Matcri

1000

Sterilization Aseptic

Spherica lParti i :les

Partiqulatg

Moi lei F(

Fo )di

Vod

Upii

10000Re

100 000

Figure 9.10 Heat transfer factory'// versus Reynolds number Re for sterilization asepticprocess and various materials.


0.1

JH 0.01

0.001100

\Part

Ficulat? Lii

oils

Sterilization Retort

1000 10000Re

100000 1000000

Figure 9.11 Heat transfer factory// versus Reynolds number Re for sterilization retortprocess and various materials.

390 Chapter 9

JH

0.001

0.0001

0.01

10 100 1000 10000 100000 1 0 0 0 0 0 0Re

Figure 9.12 Estimated equations of heat transfer factory'// versus Reynolds number Re forall the examined processes.


VI. MASS TRANSFER COEFFICIENTS IN FOOD PROCESSING:COMPILATION OF LITERATURE DATA

Recently reported mass transfer coefficient data in food processing were re-trieved from literature following the same procedure described in Section V forheat transfer coefficient data (Krokida et al., 2001b).

A total number of 15 papers were retrieved from the above journals. Thedata refer to 4 different processes (Table 9.12) and include about 9 food materials(Table 9.13). All available empirical equations were transformed in the form ofmass transfer factor versus Reynolds number (JM = aRen).

The results are classified by process and material and are presented in Ta-bles 9.14 and 9.15. All the equations are presented in Figure 9.13 to define therange of variation of the jM and Re. The range of variation by process is sketchedin Figure 9.14. The above results are presented for convective drying process inFigure 9.15. The effect of food material is obvious in this diagram. The results offitting the equation to all data for each process is summarized in Table 9.14 and inFigure 9.16.

Mass transfer coefficient values for process design can be obtained easilyform the proposed equations and graphs. The range of variation of this uncertaincoefficient can also be obtained in order to carry out valuable process sensitivityanalysis.

Table 9.12 Number of Available Equations for each Food Process

_____Process____________No. of Equations1 Drying

Convective 6Spray 1

2 FreezingForced Convection 6

3 StorageForced Convection 1

4 Sterilization______Forced Convection________________1^_____Total No. of Equations__________15

392 Chapter 9

Table 9.13 Number of Available Equations for each Food Material

123456789

MaterialComGrapesMaizeMeatModel foodPotatoesRiceCarrotsMilk

Total No. of Equations

No. of Equations121611111

15

Table 9.14 Parameters of the Equation/^ = aRe" for each Process

Process a n mm Re max Re

Drying/convective 23.5 -0.882 5 5,000

Drying/spray 2.95 -0.889 1 2

Freezing 0.10 -0.268 2,500 70,000

Storage 0.67 -0.427 50 55

Sterilization 11.2 -1.039 6,500 26,000


Table 9.15 Parameters of the Equation jM = aRe" for each Process and each Ma-terial

Process/product/reference a /; min Re max Re

Drying

ConvectiveCorn

Torrezetal., 1998 5.15Graves

Ghiausetal., 1997 0.004Vagenasetal, 1990 0.741

MaizeMouradetal., 1997 34.6

RiceTorrezetal., 1998 5.15

CarrotMuletetal., 1987 0.69

SprayMilk

Straatsma et al., 1999 2.947

FreezingMeat

Toccietal., 1995 2.496

StoragePotatoes

Xuetal.,1999 0.667

SterilizationModel food

Fuetal.,1998 11.220

-0.575 20 1,000

-0.462 10 40-0.430 900 3,000

-1.000 5 15

-0.575 20 1,000

-0.486 500 5,000

-0.890 1 2

-0.495 2,500 70,000

-0.428 50 55

-1.039 6,500 26,000

394 Chapter 9

10

0.1

JM

0.01

0.001

\\Sr—"*

%\

^

s,,

JM-1.

1

V

V

vN

N^

t,""•

llRe'0'

\-

5fcs»

ss

10

k

' Vs

x•

s

^

S

N.;

100Re

*

., »' "V1s— S

^_ _ _

^iir-5*!l«^s>

. , __

\S

\

sS

V

*;-__ 5 _ *s<v*','^n

1 000 10 000 100 0000.0001

Figure 9.13 Mass transfer factory^ versus Reynolds number Re for all the examinedprocesses and materials.


10

\

0.1

JM

0.01

0.001

0.0001

\

Con-wcltiTJfyfitg

izat

nezjir

10 100 1000Re

10 000 100 000

Figure 9,14 Ranges of variation of the mass transfer factor JM versus Reynolds numberRe for all the examined processes.

396 Chapter 9

10

0.1

JM

0.01

0.001

\Miize

Convective Drying

5

10 100Re

1000

Gra«s

10000

Figure 9.15 Mass transfer factor jM versus Reynolds number Re for convective dryingprocess and for various materials.


10

0.1

JM

0.01

0.001

\s s T a

1

1i

11 1

I>

1 1

1

tI\ing H

St nv>r

It-J^ .

1

= a

I

10

1

II BCtlV

)i ing

s, ——V\

i

5

\

j

s.ST

: ^1i j

j

1

i

^>_ . ——— ——

\\j

VS

1

1

*"*"-F

V"\X

..tod

•e=f-izil

y

\'n ttrJ

1

1J

2(

100 1000 10000 100000Re

0.0001

Figure 9.16 Estimated equations of mass transfer factory^ versus Reynolds number Refor all the examined processes.

398 Chapter 9

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Appendix: NotationA transport area, ma constant of Redlich-Kwong Eq. (2-12)Bi Biot numberb constant of Redlich-Kwong Eq. (2-12)cp heat capacity at constant pressure kJ/kmol Kcv heat capacity at constant volume kJ/kmol KC concentration, kg/rrr3

D mass diffusivity, m2/sD diameter, md diameter, mDe Deborah numberDPM dipole moment, debyeE modulus of elasticity, PaE activation energy, kJ/kmolED energy of activation for diffusion, kJ/molEa activation energy for viscous flow, kJ/molF force, NFo Fourier numberG shear modulus, PaG' storage modulus, PaG " loss modulus, PaG mass flow rate, kg/m2sGz Graetz numberh height, mh heat transfer coefficient, W/m KJA mass flux of A, kg/m2s or kmol/m2sJH heat transfer factorJM mass transfer factorK flow consistency coefficient, Pa sn

K drying constant, 1/sKp partition coefficientkB Boltzmann constant, kB= R/N= 1.38xlO"23 J/molecule Kkc mass transfer coefficient, m/sL length, mM mass, kgM torque, N mM molecular weight, kg/kmolMA molecular weight of AN Avogadro's number, 6.022xl023 molecules/molNu Nusselt numbern flow behavior index

403

404 Appendix: Notation

nPPPMPrQQqrRRertr05ShtTfTgUMu(r)VVWWVTRXX

GreekaaYF

YYSS"APs

indexpressure, Pa or barpermeability, kg / m s Papermeance, kg/ m2s PaPrandtl numbervolumetric flow rate, m3/saccumulated quantity, kg/m2

heat transport rate, Wradius, mgas constant, 8.314 kJ/kmol KReynolds numberinside radius, moutside radius, msolubility, kg/m3PaSherwood numbertime, stemperature, K, CkBT/sglass transition temperature, K, °Cvelocity, m/svelocity, m/spotential energy (Lennard-Jones potential), Jmolar volume, cm3/mol, m3/molvolume, m3

weight, kgwater vapor transmission rate, kg/m stransport propertymoisture content, kg/kg dmcompressibility factor

thermal diffusivity, m2/srelative volatilityactivity coefficientfilm flow rate, kg/m s3.141

shear rate, 1/sstrain (relative deformation)generalized transport coefficientdimensionless dipole momentpressure drop, Painteraction energy parameter, J

Appendix: Notation 405

s porosityrj viscosity, Pa srj shear viscosity, Pa s77' dynamic viscosity, Pa sT;,, apparent viscosity, Pa s77,. relative viscosity9 collision frequency, 1/s9 angle of cone/plate/I thermal conductivity, W/m KAm mean free path, mju chemical potential, kJ/molv momentum diffusivity (kinematic viscosity), m2/sp density kg/m3 or mol/m3

<r collision diameter, mT tortuosityr shear stress, PaTO yield stress, Pa<j) interaction parameter<j> volume fractionW generalized transport rateco acentric factoror frequency of oscillation. 1/s/2 collision integral/3 rotational velocity, 1/s

SubscriptsABbcDeGidKL0

PrresV

component A(diffusant)component B (medium)boilingcriticaldiffusionequilibriumgasidealKnudsenliquiddilute, initialparticlereducedresidualviscosity

Index

Absorption, 228, 234, 360Acentric factor, 11-12Acetic acid, 242, 267Activation energy, 72, 82-84, 91, 95,

198,249,253Adsorption, 113,228,231,360Agar, 298, 299, 309, 312, 321, 324Agitated kettle, 370, 371Albumen, 313, 325Albumin, 184, 193Albumine, 313, 325Almond, 175, 185, 194,226Amioca, see AmylopectinAmmonia, 241Amylopectin (Amioca), 128-132,

140,184, 193,201-204,308,320, 342

Amylose (Hylon), 128-132, 134,137,154,184,193,201-204,308, 320, 343

Apparent viscosity, 69,79, 370Apparent density, 50, 55Apple, 50-56, 149-151,182, 191,

201,213,253,275,305,317,329, 337, 377

Apple juice, 80-82, 373Apple sauce, 72, 89, 92, 94Apple slices, 253Apricots, 157, 172, 182, 191, 377Arrhenius equation, 16, 71, 78-80,

82,93, 128, 141,245,260Aseptic processing,369, 375, 381

Ash, 281Avocado, 172, 182, 191

BBaking, 373, 377, 381Baking ovens, 369Banana, 50-56, 182, 191, 201, 215,

306,318Barley, 179, 188, 302, 314, 378Beef, 183, 192, 275, 307, 319, 329,

348, 354, 379Beef carcass, 183, 192Beer, 84Beeswax, 263Beet, 186, 195,300,311,323Bentonite, 298, 355Bilberry, 296, 306, 318Bingham plastic, 68-69, 91Biopolymers, 30, 150, 254, 372Biot number, 145, 273, 363Bird-Carreau model, 73Biscuit, 168, 179, 188,302,314Blanching, 360, 375,377, 381Bluebenies, 172, 182, 191Boiling water, 369Boltzmann constant, 9, 17, 241Boundary layer theory, 368Bread, 156, 168, 179, 188, 293, 302,

314Brine, 252Broad bean, 173, 183, 192Broccoli, 176, 186, 195Broiled meat, 173, 183, 192Brown rice, 168,169, 179,180,

188-189Bull, 174, 183, 192

407

408 Index

Buoyant Force, 46Butanol, 255Butanone, 255Butter, 92, 294, 304, 316Buttermilk, 91,101Butyric acid, 242

cCake, 295,302, 314, 377Calamari, 305, 317Calcium alginate gel, 376, 379Canned foods, 369, 371Canola, 185, 194,378Capillary tube vicometer, 74-77Carbohydrate, 281Carbon dioxide, 12-13, 20, 241Carp, 295,305, 317Carrageenan gum, 85-86Carrot, 50-56, 186, 195, 201, 217,

310,322,378,393Cassava, 186, 195,310,322Catfish, 181Celery, 300,311,323Cellophane, 264Cellulose gum, 298, 309, 321Cellulose-oil-water, 174, 184, 193Cereal products, 168, 179, 188,201,

293,302,314,329Chapman-Enskog equation, 12, 15Cheese, 171, 181, 190,227,230,

251,303,315Chemical potential, 105, 109,146-

147, 244Cherry, 83, 306, 318Chicken, 174, 231, 297, 307, 319Chlorine, 241Chocolate, 70, 92, 100, 175, 185,

194,226,262-264Chromatographic method, 110, 113Clustering of solutes, 247Coating, 258-260, 262-264Cocoa, 79Coconut, 182, 191,312,324Cod, 181, 190,304,316

Codfish fillet, 145-146, 157Coffee, 194,253,312,324Colburn analogy, 368, 369Collapse, 31,50, 60, 152,259Collision, 9, 12, 13, 15Colloid / Surface Chemistry, 31Compressibility factor, 11Condensing water vapor, 364, 369Condensing water vapors, 364Controlled release, 258Cookie, 156, 168, 179, 188Cooling, 3, 227, 245, 350-351, 359,

371,375,377,381,384Corn, 156, 179, 188, 201, 209-210,

221-224, 264, 302-303, 314-315,329, 346, 378, 380, 393,

Corn cream, 376, 380Corn meal, 303,315Corn oil, 79Corn pericarp, 156, 234, 264Corn Starch, 127, 162, 174, 184,

193,353,376,378Cottonseed oil, 253Crackers, 179, 188Cracks, 47, 114, 127-128, 130, 141,

143,253,260Cream, 92, 303-304, 315-316Critical conditions, 11-12,14-15Crystallization, 30, 33, 360Cucumber, 301,311, 323

&Dairy products, 181, 190,303,315,

329Data banks, 7, 19, 29, 269, 275Database, 27, 144, 161, 164Deborah number, 73, 108Diaphragm cell, 238-239Dickerson method, 273, 351Dietary fibers, 89Diethyl ketone, 255Diffusivity, mass

determination, 109-123sorption kinetics, 110

Index 409

[Diffusivity, mass]permeability methods, 114distribution of diffusant, 118drying methods, 120flavors, 254-258in fluid foods, 241-243in polymers. 243organic components, 252salts, 251small solutes, 237

Diffusivity, moisturebaked products, 168, 179, 188cereal products, 168,179,188, 201,

209-212dairy products, 171,181, 190fish, 171,181, 190fruits, 172, 182, 191, 201, 205, 206legumes, 173, 183, 192meat, 173, 183, 192model foods,127-143,174, 184,

193,201,203,204nuts, 175, 185, 194other, 175, 185, 194vegetables, 176, 186, 195,201

Dilatant fluid, 69Dimensional equations, 363Dimensionless numbers, 14, 362,

363,366-367,371,374Distillation, 266, 360Distribution of diffusant, 110, 118Dogfish, 171, 181, 190Dough, 168, 179, 188,229,293,

302,314,329,347,352-353,356

Dried fruit, 275Dry milk, 181,190Dry solids apparent density, 55Dry solids true density, 55Drying Kinetics, 110, 159-160, 162,

227, 230-232, 234Dual-sorption model, 248, 262Dynamic viscosity, 73

Edible oils, 72, 78-79, 92-93Egg, 175, 185, 194,229Einstein equation, 66Elastic materials, 65, 244Empirical rules, 279Emulsions, 90, 92, 262Enclosed water density, 50, 55Enzyme inactivation, 360Ethanol, 12, 13, 17, 20, 84, 241-242,

255Ethyl acetate, 255Ethyl butyrate, 255Ethyl oleate, 153, 154Ethylene, 12-13, 20, 241Eucken factor, 15Evaporation, 254-257, 360, 372Evaporators, 369Excess contributions, 14Extensional viscosity, 64Extraction, 33-34, 252-253, 360Extrusion, 130-131Eyring theory, 16, 18

Fababean, 173, 183, 192Falling films, 364, 367Falling rate period, 120-122, 144-

145,369Fat, 232-233, 281, 304, 307, 316,

319,355Fiber, 101,281Fibrinogen, 242Pick diffusion equation, 8, 106, 118-

119,145-147,237,243Fickian diffusion, 107-108, 243,

250,260Figs, 376-377Fish, 122, 145,157,181,190,252,

275,304,316,379Flavor, 254-259, 261-262Flavor encapsulation, 258

410 Index

Flour, 184, 193, 302-303, 314-315,324

Flow behavior index, 68, 72, 76, 78,80, 85, 88-92, 95

Flow consistency coefficient, 68,370-371

Fluidisedbeds, 375, 378, 381Food coatings, 262-263Food Materials Science, 2, 30Food preparation, 360Food preservation, 360Food rheology, 3, 63Free convection, 363Free-volume model, 249Freeze-dried gel, 275Freezing, 379,381,391-393Fructose, 242, 266Fruits, 50, 94, 146-152,157, 182,

191,201,205-206,305-306,317-318,329,331-332,373

Frying, 35, 360, 369GGamma function, 125Garlic, 186, 195, 201, 219, 310,322,Gas bubbles, 367Gas constant, 16, 71, 93, 198, 200,

249, 326, 328Gas pycnometer method, 46Gelatin, 281, 298, 309-310, 321-322Gelatinized starch,130, 141, 275,

281Gelatin, 298, 309-310, 321-322Glass transition, 30-31, 244-247Glucose, 242, 312, 324Gluten, 174, 184, 193,263-264Glycerin, 242, 263, 298, 310, 322Glycine, 242Graetz number, 363, 369Granular materials, 36, 61, 133, 135,

285Granular starch, 130, 133-135, 137-

141, 150, 154,275,284

Grapes, 191,201,296,306,318,377-378, 393

Grashof number, 363Gravimetric method, 110Green beans, 376-377Green olives 251Ground beef, 174, 183, 192Guar gum, 85-86Guarded hot plate, 270-271

HHaddock, 171, 181, 190Halibut, 171, 182, 191Hamburger, 303, 315, 376, 379Hazelnuts, 175, 185, 194HOPE, 261-262, 264Heat capacity, 8, 15Heat exchangers, 369Heat transfer coefficients,

determination, 361baking, 375, 377, 381blanching, 375, 377, 381cooling, 375, 377, 381,384drying, 375, 378, 381,385freezing, 375, 379,381,387storage, 375, 379, 381, 387sterilization, 375, 380, 381, 388-

389Heat transfer correlations, 362-363,

371Heat transfer factor, 363, 374, 382,

384-389Heated probe, 270-274Heifer, 184, 193Hemoglobin, 242Herring, 157, 182,191,251Herschel-Bulkley equation, 68-69,

75,91,95Hexane, 253Hexanol, 255Honey, 78, 80, 83, 299, 313, 325Horizontal tubes, 364Huggins equation, 65Hydrodynamic flow, 147

Index 411

Hydrogen, 241Hylon, see Amylose

IIce, 33, 47, 276, 281, 292, 325, 327Idli Batter, 293, 303, 315

JJuices, 80-83, 89-90, 92, 95-99, 305-

306,311,317-318,323,373

KKaraya gum, 85, 87Kidney bean, 301,311,323Kinematic viscosity, 8, 362

LLactoglobulin, 242Lard, 299, 312, 324LDPE, 261-262, 264Legumes, 173, 183, 192, 296, 306Lentils, 183, 192,306,318Lewis number, 363Lewis-Squires equation, 16, 72Licorice extract, 83Lipid films, 264Liquid diffusion, 141, 143Liquid displacement method, 46

MMacadamia, 298, 310, 322Mackerel, 157, 182, 191, 304, 316Macrostructure, 2-4, 29, 35, 40, 45-

47, 49-50Maize, 303, 315, 378, 393Malt, 180, 189,230,376,378Maltose, 242Mango, 89, 90, 92, 95, 99, 101-102Margarine, 33, 73, 92Mass transfer coefficients

determination, 365drying, 392-394, 397, 397freezing, 391-393, 397storage, 391-393, 397sterilization, 391-393, 397

Mass transfer correlations, 359, 362,366-367

Mass transfer factor, 363, 368, 391,395-398

Mass transfer operations, 109, 359-360, 362

Mayonnaise, 90, 92Meat, 157,183, 192, 251, 307, 319,

329, 379, 393Meat Carcass, 376, 379Meat Muscle, 251Meatballs, 376, 379Membrane, 366Methane, 241Methanol, 255Method of slopes, 123-124Methyl anthranilate, 255Microstructure, 2, 5, 29, 31-35, 60,

114,243,261Milk, 90-91, 181, 190, 275-276,

304,312,316,324,329,345,393

Milled Rice, 169, 180, 189Mixed model, 283-284Mizrahi-Berk model, 70Model food, 346-49,142, 203Molecular diameter, 9, 12Molecular diffusion, 4, 106-107,

144, 147, 238, 244, 260, 263Molecular effusion, 107Molecular simulation, 2, 5,29- 30,

61,248,250,253,268Mulberry, 182, 191,231Mushrooms, 301, 311, 323, 357Mustard, 92Mutton, 297, 308, 320

NNatural convection, 363-364, 369Navy beans, 183, 192,232Nernst-Haskel equation, 18Newton equation, 8, 64, 65, 73Newtonian foods, 69, 72, 380Nitrogen, 12-13,20,241

412 Index

Non-Newtonian foods, 68, 372Nuclear magnetic resonance, 75Numerical methods, 124-125Nusselt number, 351, 363, 399Nuts, 175,185, 194, 298, 310, 322

oOat, 294, 303, 315Okra, 176, 186, 195,228Olive oil, 79Onion, 156, 186, 195, 201, 218, 310,

322Orange, 83, 89, 95, 97, 306, 318,

329,338Osmotic dehydration, 35, 147Ovalbumin, 242, 281, 309, 321Overall heat transfer coefficient,

371-373Oxygen, 12-13,20,241,262

PPackaging, 259-264Packed beds, 366Paddy Rice, 180, 189Palm kernel, 299, 312, 324Paprika, 176, 186, 195, 227Parallel flow, 364Parallel model, 65, 283, 284Parboiled Rice, 169, 180, 189,227Parsley, 300, 311,323Parsnip, 301,311,323Past single sphere, 363, 366Pasta, 156, 169, 180, 189, 201, 225Pasteurization, 360Peas, 177, 186, 195, 300, 310, 322Peaches, 183, 192, 306, 318, 377Peanuts, 175, 185, 194,253Peanut butter, 92Peanut oil, 253Peanut pods, 175, 185, 194,227Pear, 95, 98, 102, 306, 318, 329, 339Peclet number, 363Pectin, 33, 88-89, 127, 256-257,

298,309-310,321-322

Penetration theory, 367Pepper, 177, 186, 195Pepperoni, 174, 184, 193, 308, 320Permeability, 110, 114-117,237-

238, 243-244, 248-249, 259-264Permeance, 260Perpendicular flow, 364Phase transition, 30, 61Pickles, 251,267Pigeon pea, 177, 186, 195,233Pineapple, 173, 183, 192, 226, 232Plant cells, 32-33Plant hydrocolloids, 85Plantain, 306, 318Plasticization, 128, 244Poiseuille equation, 74Polyacrylamide gel, 175, 184, 193Polymer Science, 30, 65,110, 244Polysaccharide films, 264Pork, 157, 265, 297, 308, 320, 355Potatoes, 50, 54, 150, 151, 156, 184,

186,193, 195,201,216,275,297,310,322,329,340,378-379, 392-393

Potato starch, 158, 175, 184, 193,297, 309, 321

Power-law model, 68-69, 71, 75, 85,88-89, 91, 94-95

Prandtl number, 363Propanol, 255Protein films, 264Proteins, 135-136, 155,262-264,

280-281Pseudoplastic fluids, 69, 85, 88-89,

252Puffing, 127,152,231,275PVC, 261-262, 264

RRadish, 301,311,323Raisins, 157, 183, 192, 201, 306 318Rapeseed, 312, 324, 329Raspberries, 376-377Raspberry, 89, 102, 295, 306, 318

Index 413

Real gases, 11-12, 15Recovery of volatiles, 360Refrigeration, 360, 369Regular regime theory, 125-126Retorts, 375, 380-381Reverse Osmosis, 360Rheological properties (viscosity)

aqueous Newtonian, 80chocolate, 100cloudy juices/pulps, 89edible oils, 79emulsions, 90fruit and vegetable juices, 95plant biopolymer solutions, 85

Rheology, 3, 5, 63, 66, 70, 72-74,90, 101-103

Rice, 156, 179-180, 185, 188-189,201,303,315,378,393

Rice Starch, 185, 194Rotary drying, 375, 379, 381Rotational viscometer, 75-76Rough Rice, 180, 189, 201, 233Rubbery state, 108, 238, 244Rutabagas, 301,311

5Salad, 90Salmon, 91, 101,295,305,317Salt, 18, 20, 84, 133, 146-147, 237,

251-252Sausage, 157,184, 193, 308, 320Scanning microscopes, 31-32Schmidt number, 363Shark, 171, 182, 191Shear rate, 31, 64-66, 68-76, 78-80,

85,88,91,94,370Shear stress, 8, 64-66, 68-71 74-76,

78, 85, 88, 94Sherwood number, 120-121, 126,

161,363,365Shrimp, 305,317Shrinkage, 36-37, 39-40, 47, 50, 55,

60, 126, 137, 150Shrinkage coefficient, 55

Simplified methods, 110, 123, 125Simulation, 29-30, 124, 232, 250,

253Skim milk, 101, 171, 181, 190, 352Sodium caseinate, 101, 263Solid displacement method, 46Solubility, 115, 117-118,248-249,

259-260, 262-263Sorbitol, 263Sorghum, 299, 313, 325Sorption kinetics, 107-111, 127,

129, 148,150-152,244Soy flour, 293, 302, 314Soya, 177, 186, 195,376,379Soya meal, 177, 186, 195Soybean, 152, 156, 186, 195,312,

324, 378Soybean flakes, 253Soybean oil, 79,253Soybeans, 150, 152, 156,161Specific volume, 55Spinach, 301, 311,323, 351Spinning disc, 366Squid, 191,305,317Stanton number, 363Starch gel, 49, 127-129, 131Steam, 19, 27, 226, 352, 375Sterilization, 360, 380-381, 388-

389, 392-394Stokes diaphragm cell, 239Stokes-Einstein equation, 17,241,

242, 252Storage, 30-32, 257, 379, 381 387,

391-393Strawberries, 295, 306, 318, 351,

376-377Structural models, 40, 47, 50, 163,

197,280,283,284,287,326Structural properties, 55Sucrose, 24-25, 80-83, 133, 147-

148,242,253,258,281,309-310,312,321-322,324,371

Sucrose solution, 275

414 Index

Sugar Beets, 178, 186, 195,253,300,311,323

Sugar solutions, 80Sunflower seeds, 185, 194Surface renewal theory, 367Surfactants, 153Surimi, 305, 317Swiss cheese, 251,267Swordfish, 172, 182, 191

TTailor-made porous solid foods, 47,

60Tapioca, 178, 186, 195Taylor dispersion method, 238, 240Texture, 3, 29, 31, 33, 60, 64, 101,

102-104,266Thermal conductivity of foodsdetermination, 270-273baked products, 293, 302, 314, 329cereal products, 293, 302, 314, 329dairy products, 294, 303, 314, 329fish, 294, 304, 316fruits, 295, 305, 317, 329, 331-332legumes, 296, 306, 318meat, 296, 307, 319,329model foods, 297, 308, 320, 329nuts, 298, 310, 322other, 299, 312, 324, 329vegetables, 300, 310, 322, 329,

333-334Thermal diffusion, 107, 145Thermal diffusivity, 269, 273-279Thermodynamics, 7, 10, 27, 29, 158,

266-267Thixotropic fluids, 70-71, 91Time lag method, 110, 117Tobacco, 299, 312,324, 350Tomato, 67-68, 90, 95-96, 187, 196,

252,311,323,329,341,377Tomato paste, 275Torque, 74, 76Tortuosity, 106, 147, 250, 254

Transport coefficients, 144Transport gradient, 8Transport Phenomena, 5, 7, 27, 158,

228, 257, 265, 354True density, 55Turbulent flow, 363, 366, 369Turkey, 184, 308, 320Turnip, 187, 196,311,323Tylose, 309,321

uUnsteady-state method, 110, 117-

118,273-274

VVapor diffusion, 106, 109, 141, 143,

163, 199Variable Diffusivity, 110, 123, 226Vegetable oil, 275Vegetables, 35, 50, 95, 150, 156,

176,186, 195,201,207-208,333-334

Vertical tubes, 364Viscometers

capillary tube, 74rotational, 75cone-and-plate, 76

Viscosity, see Rheological proper-ties

Volatile compounds, 255, 266Volume displacement method, 46

wWeissenberger rheogoniometer, 76Wheat, 135, 156, 181, 190,201,

220,302-303,315,378-379Whey, 263, 294, 304,316Whiting, 172, 182, 191Wild rice, 170, 181, 190Wilke-Chang equation, 241-242,

252Williams-Landel-Ferry equation, 31,

245Wine, 84, 102

Index 415

x yXanthan gum, 85, 87 Yam, 187, 196, 311, 323, 355

Yellow batter, 293Yogurt, 91, 101, 294, 304, 316, 353